<PAGE>   1
                          UNITED STATES
               SECURITIES AND EXCHANGE COMMISSION
                    Washington,  D.C.  20549



                            FORM 10-Q
       Quarterly Report Pursuant to Section 13 or 15(d) of
               the Securities Exchange Act of 1934


For the Quarter Ended June 30, 1998  Commission File No. 1-5591


                        PENNZOIL COMPANY
     (Exact name of registrant as specified in its charter)


             Delaware                          74-1597290
 (State or other jurisdiction of             (I.R.S Employer
  incorporation or organization)             Identification No.)


                  Pennzoil Place, P.O. Box 2967
                   Houston, Texas  77252-2967
             (Address of principal executive offices)



Registrant's telephone number, including area code: (713) 546-4000


      Indicate by check mark whether the registrant (1) has filed all reports
required to be filed by Section 13 or 15(d) of the Securities Exchange Act of
1934 during the preceding 12 months (or such shorter period that the registrant
was required to file such reports), and (2) has been subject to such filing
requirements for the past 90 days. Yes X . No .

      Number of shares outstanding of each class of stock, as of latest
practicable date, July 31, 1998:

     Preferred stock, par value $1.00 per share, 1,500,000 shares.

     Common stock, par value $0.83-1/3 per share, 47,755,293 shares.

<PAGE>   2
                          PART I. FINANCIAL INFORMATION

Item 1. Financial Statements
----------------------------

<TABLE>
                                                     PENNZOIL COMPANY
                          CONDENSED CONSOLIDATED STATEMENT OF INCOME AND COMPREHENSIVE INCOME
                                                        (UNAUDITED)
<CAPTION>

                                                                       Three Months Ended                  Six Months Ended
                                                                             June 30                           June 30
                                                                  ----------------------------      ----------------------------
                                                                     1998             1997             1998             1997
                                                                  -----------      -----------      -----------      -----------
                                                                        (Expressed in thousands except per share amounts)
<S>                                                               <C>              <C>              <C>              <C>
REVENUES                                                          $  632,503       $  648,357       $1,178,313       $1,297,357

COSTS AND EXPENSES
   Cost of sales                                                     381,214          373,293          688,996          729,165
   Selling, general and administrative                                92,885           98,111          179,574          178,213
   Depreciation, depletion and amortization                           73,297           74,784          146,659          137,351
   Exploration                                                        19,220           12,279           31,895           22,319
   Taxes, other than income                                           10,478           11,751           21,942           24,863
   Interest charges, net                                              43,499           41,161           85,379           78,280
                                                                  -----------      -----------      -----------      -----------
INCOME BEFORE INCOME TAX                                              11,910           36,978           23,868          127,166

Income tax provision                                                   3,083           13,101            5,383           45,738
                                                                  -----------      -----------      -----------      -----------
NET INCOME                                                             8,827           23,877           18,485           81,428
Preferred stock dividend                                                 757             -                 757             -
                                                                  -----------      -----------      -----------      -----------
NET INCOME AVAILABLE TO
   COMMON SHAREHOLDERS                                            $    8,070       $   23,877       $   17,728       $   81,428
                                                                  ===========      ===========      ===========      ===========

NET INCOME                                                        $    8,827       $   23,877       $   18,485       $   81,428
OTHER COMPREHENSIVE INCOME
   Foreign currency translation adjustment                            (2,142)          (2,653)          (3,017)          (2,652)
   Unrealized gains on securities, net of tax                         (1,902)             (22)          (1,501)          (1,041)
                                                                  -----------      -----------      -----------      -----------
OTHER COMPREHENSIVE INCOME, NET OF TAX                                (4,044)          (2,675)          (4,518)          (3,693)
                                                                  -----------      -----------      -----------      -----------
COMPREHENSIVE INCOME                                              $    4,783       $   21,202       $   13,967       $   77,735
                                                                  ===========      ===========      ===========      ===========


EARNINGS PER SHARE
   Basic                                                          $     0.17       $     0.51       $     0.37       $     1.74
   Diluted                                                        $     0.17       $     0.50       $     0.37       $     1.71


DIVIDENDS PER COMMON SHARE                                        $     0.25       $     0.25       $     0.50       $     0.50
                                                                  ===========      ===========      ===========      ===========

AVERAGE SHARES OUTSTANDING
   Basic                                                              47,695           46,971           47,643           46,888
                                                                  ===========      ===========      ===========      ===========
   Diluted                                                            48,320           47,450           48,359           47,446
                                                                  ===========      ===========      ===========      ===========

END OF PERIOD SHARES OUTSTANDING                                      47,727           47,031           47,727           47,031
                                                                  ===========      ===========      ===========      ===========
<FN>
<F1>See Notes to Condensed Consolidated Financial Statements.
</FN>
</TABLE>

<PAGE>   3
                   PART I. FINANCIAL INFORMATION - continued

<TABLE>
                                               PENNZOIL COMPANY
                                    CONDENSED CONSOLIDATED BALANCE SHEET
                                                 (UNAUDITED)
<CAPTION>
                                                                               June 30,           December 31,
                                                                                1998                  1997
                                                                            -------------        -------------
                                                                                 (Expressed in thousands)
<S>                                                                         <C>                   <C>
ASSETS

Current assets
   Cash and cash equivalents                                                $      20,971        $      18,594
   Receivables                                                                    245,680              234,282
   Inventories
     Crude oil and natural gas                                                     18,900               20,883
     Motor oil and refined products                                               183,139              184,027
   Deferred income tax                                                             19,691               19,479
   Other current assets                                                            76,840              117,449
                                                                            -------------        -------------
Total current assets                                                              565,221              594,714

Property, plant and equipment, net                                              2,591,061            2,498,597
Marketable securities and other investments                                       947,301              945,995
Other assets                                                                      358,199              366,581
                                                                            -------------        -------------

TOTAL ASSETS                                                                $   4,461,782        $   4,405,887
                                                                            =============        =============

LIABILITIES AND SHAREHOLDERS' EQUITY

Current liabilities
   Current maturities of long-term debt                                     $         219        $       2,363
   Accounts payable and accrued liabilities                                       236,056              363,249
   Interest accrued                                                                28,967               30,016
   Other current liabilities                                                       61,965               90,367
                                                                            -------------        -------------
Total current liabilities                                                         327,207              485,995

Long-term debt, less current maturities
   Exchangeable debentures                                                        888,627              889,027
   Other long-term debt                                                         1,358,759            1,308,520
                                                                            -------------        -------------
Total long-term debt, less current maturities                                   2,247,386            2,197,547
Deferred income tax                                                               296,992              288,677
Other liabilities                                                                 304,784              295,129
                                                                            -------------        -------------
TOTAL LIABILITIES                                                               3,176,369            3,267,348
                                                                            -------------        -------------
COMMITMENTS AND CONTINGENCIES

SHAREHOLDERS' EQUITY                                                            1,285,413            1,138,539
                                                                            -------------        -------------

TOTAL LIABILITIES AND SHAREHOLDERS' EQUITY                                  $   4,461,782        $   4,405,887
                                                                            =============        =============
<FN>
<F1>See Notes to Condensed Consolidated Financial Statements.
</FN>
</TABLE>



<PAGE>   4
                   PART I. FINANCIAL INFORMATION - continued

<TABLE>
                                                 PENNZOIL COMPANY
                                  CONDENSED CONSOLIDATED STATEMENT OF CASH FLOWS
                                                    (UNAUDITED)
<CAPTION>

                                                                                           Six Months Ended
                                                                                                June 30
                                                                                   ---------------------------------
                                                                                      1998                  1997
                                                                                   -----------           -----------
                                                                                        (Expressed in thousands)
<S>                                                                                <C>                   <C>
CASH FLOWS FROM OPERATING ACTIVITIES
  Net income                                                                       $   18,485            $   81,428
  Adjustments to reconcile net income to net cash
    provided by operating activities:
      Depreciation, depletion and amortization                                        146,659               137,351
      Dry holes and impairments                                                        12,353                 9,231
      Deferred income tax                                                               4,696                28,681
      Equity investee distributions in excess of earnings                               4,925                  -
      Non-cash and other nonoperating items                                            18,168                22,270
      Changes in operating assets and liabilities:
        Accrued taxes                                                                 (29,285)                6,315
        Accounts payable and other accrued liabilities                                (82,136)               19,592
        Other assets and liabilities                                                  (24,994)              (31,666)
                                                                                   -----------           -----------
  Net cash provided by operating activities                                            68,871               273,202
                                                                                   -----------           -----------

CASH FLOWS FROM INVESTING ACTIVITIES
  Capital expenditures                                                               (275,724)             (254,436)
  Purchases of marketable securities and other investments                           (294,550)             (272,980)
  Proceeds from sales of marketable securities and other
    investments                                                                       292,844               277,161
  Proceeds from sales of assets                                                        48,742                10,228
  Other investing activities                                                          (12,327)              (36,778)
                                                                                   -----------           -----------
  Net cash used in investing activities                                              (241,015)             (276,805)
                                                                                   -----------           -----------

CASH FLOWS FROM FINANCING ACTIVITIES
  Proceeds from notes payable, net                                                     82,342                43,622
  Debt and capital lease obligation repayments                                       (827,643)             (773,480)
  Proceeds from issuance of debt                                                      791,457               750,000
  Net proceeds from issuance of preferred stock                                       147,000                  -
  Dividends paid                                                                      (23,828)              (23,465)
  Other financing activities                                                            5,193                16,428
                                                                                   -----------           -----------
  Net cash provided by financing activities                                           174,521                13,105
                                                                                   -----------           -----------

NET INCREASE IN CASH AND CASH EQUIVALENTS                                               2,377                 9,502


CASH AND CASH EQUIVALENTS, beginning of period                                         18,594                34,383
                                                                                   -----------           -----------

CASH AND CASH EQUIVALENTS, end of period                                           $   20,971            $   43,885
                                                                                   ===========           ===========
<FN>
<F1>See Notes to Condensed Consolidated Financial Statements.
</FN>
</TABLE>


<PAGE>   5

           PART I. FINANCIAL INFORMATION - continued

                         PENNZOIL COMPANY
       NOTES TO CONDENSED CONSOLIDATED FINANCIAL STATEMENTS
                            (UNAUDITED)


(1)  General -

     The condensed consolidated financial statements included herein have been
prepared by Pennzoil Company ("Pennzoil") without audit and should be read in
conjunction with the financial statements and the notes thereto included in
Pennzoil's latest annual report. The foregoing financial statements include only
normal recurring accruals and all adjustments which Pennzoil considers necessary
for a fair presentation. Certain prior period items have been reclassified in
the condensed consolidated financial statements in order to conform with the
current year presentation.

(2)  New  Accounting Standards -

     In June 1997, the Financial Accounting Standards Board ("FASB") issued
Statement of Financial Accounting Standards ("SFAS") No. 131, "Disclosure about
Segments of an Enterprise and Related Information," which establishes standards
for reporting information about operating segments in annual financial
statements and requires that selected information be reported about the
operating segments in interim financial reports issued to the shareholders. It
also establishes standards for related disclosure about products and services,
geographic areas, and major customers. Pennzoil plans to adopt SFAS No. 131 in
its annual financial statement disclosures for the fiscal year ending December
31, 1998.
     In March 1998, the American Institute of Certified Public Accountants
("AICPA") issued Statement of Position ("SOP") No. 98- 1, "Accounting for the
Costs of Computer Software Developed or Obtained for Internal Use." This SOP is
effective for fiscal years beginning after December 15, 1998 and earlier
adoption is permitted. The adoption of SOP No. 98-1 is not expected to have a
material impact on Pennzoil's results of operations.
     In April 1998, the AICPA issued SOP No. 98-5, "Reporting on the Costs of
Start-Up Activities." This SOP is effective for financial statements for fiscal
years beginning after December 15, 1998 and earlier adoption is permitted.
Pennzoil is currently evaluating the impact of SOP No. 98-5.
     In June 1998, the FASB issued SFAS No. 133, "Accounting for Derivative
Instruments and Hedging Activities." The SFAS establishes accounting and
reporting standards requiring that every derivative instrument be recorded in
the balance sheet as either an asset or liability measured at its fair value.
The SFAS requires that changes in the derivative's fair value be recognized
currently in earnings unless specific hedge accounting criteria are met.
Accounting for qualifying hedges allows a derivative's gains and losses to
offset related results on the hedged item in the income statement, and requires
a company to formally document, designate, and assess the effectiveness of
transactions that receive hedge accounting. SFAS No. 133 is effective for fiscal
years beginning after June 15, 1999 and early adoption is permitted. The effect
of adopting SFAS No. 133 has not been determined, but is not expected to have a
material impact on Pennzoil's results of operations (Reference is made to Note 5
of Notes to Condensed Consolidated Financial Statements).

<PAGE>   6

           PART I. FINANCIAL INFORMATION - continued

(3)  Proposed Spin-off of Downstream Operations and  Merger
     of Downstream Operations with Quaker State Corporation -

     On April 14, 1998, Pennzoil, Pennzoil's subsidiary Pennzoil Products
Company ("PPC") and Downstream Merger Company, a wholly owned subsidiary of PPC
("Merger Sub"), entered into an Agreement and Plan of Merger (the "Merger
Agreement") with Quaker State Corporation ("Quaker State"). The Merger Agreement
and related agreements provide for the separation of Pennzoil's motor oil,
refined products and fast lube operations (which generally includes PPC and
Jiffy Lube International, Inc. ("Jiffy Lube") and their respective subsidiaries)
from its exploration and production operations and for the combination of the
motor oil, refined products and fast lube operations with Quaker State.
     The transactions contemplated by the Merger Agreement are (1) a pro rata
distribution, on a share-for-share basis, of all of the issued and outstanding
Common Stock of PPC (which, among other things, will at such time hold the motor
oil and refined products operations of PPC and the fast lube operations of Jiffy
Lube) to the holders of Common Stock of Pennzoil and (2) a merger of Merger Sub
with and into Quaker State, in which holders of Capital Stock of Quaker State
will receive, in exchange for each share held, 0.8204 shares of Common Stock of
PPC. Immediately following the transactions contemplated by the Merger
Agreement, approximately 38.5% of PPC will be owned by former Quaker State
stockholders and approximately 61.5% of PPC will be owned by stockholders of
Pennzoil.
     Closing under the Merger Agreement is conditioned upon, among other things,
approval by Quaker State's stockholders and receipt of a favorable tax ruling
from the Internal Revenue Service. The required waiting period under the
Hart-Scott-Rodino Antitrust Improvements Act of 1976 has expired. The spin-off
and merger are expected to occur during the fourth quarter of 1998.

(4)  Debt -

     During the six months ended June 30, 1998, certain holders of Pennzoil's
exchangeable debentures exchanged debentures for Chevron Corporation ("Chevron")
common stock, in accordance with the respective supplemental indentures, which
resulted in a decrease of $.4 million of outstanding exchangeable debentures.
     The debentures are exchangeable at the option of the holders at any time
prior to maturity, unless previously redeemed, for shares of Chevron common
stock. In lieu of delivering Chevron common stock, Pennzoil may, at its option,
pay to any holder an amount in cash equal to the market value of the Chevron
common stock to satisfy the exchange request. If Pennzoil delivers Chevron
common stock to satisfy an exchange, Pennzoil records an ordinary gain on the
sale of Chevron common stock, limited to the face value of the related
exchangeable debentures and the associated debt issue cost. The face value of
exchangeable debentures and debt issue cost would be retired by recording an
extraordinary charge for the early extinguishment of debt. Alternatively, should
Pennzoil choose to pay cash to the holders and retain the Chevron common stock,
Pennzoil would record an extraordinary charge for the extinguishment of debt. In
addition, Pennzoil would record an increase in the net unrealized holding gain
on the Chevron common stock to reflect current market value which was previously
capped under the exchange rights. This would be reported as an increase in other
comprehensive income in accordance with SFAS No. 130 (Reference is made to Note
7 of Notes to Condensed Consolidated Financial Statements) and as a separate
component of shareholders' equity as required under SFAS No. 115, "Accounting
for Certain Investments in Debt and Equity Securities."

<PAGE>   7

           PART I. FINANCIAL INFORMATION - continued

     On July 1, 1998, Pennzoil commenced an offer to issue new exchangeable
senior debentures ("New Debentures") in exchange for a portion of its 6.5%
Exchangeable Senior Debentures (the "6.5% Debentures") and 4.75% Exchangeable
Senior Debentures (the "4.75% Debentures"). The exchange offer expired on July
31, 1998 and on August 3, 1998, Pennzoil issued $443.8 million principal amount
of 4.9% Debentures in exchange for $211.6 million principal amount of 6.5%
Debentures and $317.4 million principal amount of 4.95% Debentures in exchange
for $211.6 million principal amount of 4.75% Debentures. Pennzoil will realize
an extraordinary loss on early extinguishment of debt of $203.8 million, which
is the difference between the carrying amount of the related exchanged
debentures of $420.7 million (net of related unamortized debt issue costs of
$2.5 million) and the estimated market value (net of anticipated discount) of
the New Debentures being issued of $739.1 million. This pretax loss of $318.4
million is then reduced by an income tax benefit of $114.6 million.
    Effective September 7, 1998, Pennzoil will redeem the remaining $184.1
million of 6.5% Debentures. Pennzoil expects that at the current prevailing
market prices for Chevron stock, substantially all holders of 6.5% Debentures
and 4.75% Debentures will exercise their existing exchange rights to obtain
shares of Chevron stock prior to that date (although Pennzoil has the right to
pay cash instead of delivering shares of Chevron stock). Assuming that Pennzoil
elects to deliver stock, Pennzoil will realize an after-tax gain of $23.6
million, and a $.7 million extraordinary loss on the write-off of the remaining
unamortized debt issue cost related to the 6.5% Debentures. If Pennzoil elects
to pay cash, Pennzoil will realize: (1) an extraordinary loss based on the
market value of Chevron stock at the time of the exchange less the carrying
value of the debt, (2) a $.7 million extraordinary loss on the write-off of the
remaining unamortized debt issue cost, and (3) an ordinary gain on the sale of
the Chevron shares at the then current market price less the original cost to
Pennzoil of $33.68 per share.
     Pennzoil expects to redeem the remaining $279.2 million of 4.75% Debentures
on or about the first call date, October 1, 1998.
     Completion of the exchange offers allows the Company to release
approximately 1.5 million shares of Chevron stock that previously had been
allocated for the exchange rights of the 6.5% Debentures and 4.75% Debentures.
Pennzoil currently intends to sell these shares from time to time.

(5)  Use of Derivatives -

     Pennzoil has a price risk management program that utilizes derivative
financial instruments, principally crude oil and natural gas swaps, to reduce
the price risks associated with fluctuations in crude oil and natural gas
prices. These financial instruments are designated as hedges and accounted for
on the accrual basis with gains and losses being recognized based on the type of
contract and exposure being hedged. Realized gains or losses on crude oil and
natural gas swaps designated as hedges of anticipated transactions related to
anticipated production are treated as deferred credits or charges and are
included in other current liabilities or other current assets on the balance
sheet. Net gains and losses on crude oil and natural gas swaps designated as
hedges of anticipated transactions, including accrued gains or losses upon
maturity or termination of the contract, are deferred and recognized in income
when the associated hedged commodities are produced.

<PAGE>   8
           PART I. FINANCIAL INFORMATION - continued

     In order for crude oil and natural gas swaps to qualify as a hedge of an
anticipated transaction, the derivative contract must identify the expected date
of the transaction, the commodity involved, and the expected quantity to be
purchased or sold. In the event that a hedged transaction does not occur, future
gains and losses, including termination gains or losses, are included in the
income statement when incurred. In the statement of cash flows, cash receipts or
payments related to financial instruments are classified consistent with the
cash flows from the transaction being hedged. Effective January 1, 2000, the
accounting for derivative financial instruments will be amended as required
pursuant to SFAS No. 133 (Reference is made to Note 2 of the Notes to Condensed
Consolidated Financial Statements). Under this new standard, crude oil and
natural gas swaps will be marked to fair value with unrealized changes in the
fair value of the crude oil and natural gas swaps recorded as comprehensive
income. At the time of maturity or termination of the contract, any deferred
gain or loss is recognized in earnings.
     Pennzoil has not materially hedged crude oil or natural gas prices in 1998.
Pennzoil will continually review and may alter its hedged positions as
conditions change.

(6)  Earnings Per Share -

     Earnings per share of common stock outstanding were computed as follows:


<TABLE>

                                                                       Three Months Ended                  Six Months Ended
                                                                             June 30                           June 30
                                                                  ----------------------------      ----------------------------
                                                                     1998             1997             1998             1997
                                                                  -----------      -----------      -----------      -----------
                                                                        (Expressed in thousands except per share amounts)
<S>                                                               <C>              <C>              <C>              <C>
Net income                                                        $    8,827       $   23,877       $   18,485       $   81,428

Less:  Preferred stock dividend                                          757             -                 757             -

Net income available to common shareholders                       $    8,070       $   23,877       $   17,728       $   81,428

Basic weighted average shares                                         47,695           46,971           47,643           46,888

Effect of dilutive securities:<F1>
          Options                                                        478              360              569              438
          Awards                                                         147              119              147              120

Diluted weighted average shares                                       48,320           47,450           48,359           47,446

Earnings per share
  Basic                                                           $     0.17       $     0.51       $     0.37       $     1.74
  Diluted                                                         $     0.17       $     0.50       $     0.37       $     1.71

<FN>
  <F1>A weighted average number of options to purchase 1,254,291 and 1,460,756
      shares of common stock were outstanding for the three months ended June
      30, 1998 and 1997, respectively, but were not included in the computation
      of diluted earnings per share because the options' exercise prices were
      greater than the average market price of the common shares. A weighted
      average number of options to purchase 860,499 and 1,211,282 shares of
      common stock were outstanding for the six months ended June 30, 1998 and
      1997, respectively, but were not included in the computation of diluted
      earnings per share because the options' exercise prices were greater than
      the average market price of the common shares.
</FN>
</TABLE>

<PAGE>   9
           PART I. FINANCIAL INFORMATION - continued


(7)  Comprehensive Income -

     In June 1997, the FASB issued SFAS No. 130, "Reporting Comprehensive
Income," which requires that an enterprise classify items of other comprehensive
income by their nature in a financial statement and display the accumulated
balance of other comprehensive income separately from retained earnings and
additional paid-in capital in the equity section of the balance sheet. Pennzoil
adopted SFAS No. 130 in the first quarter of 1998. Components of comprehensive
income consist of foreign currency translation adjustments, unrealized holding
gains and losses on available-for-sale securities and minimum pension liability.

(8)  Preferred Stock -

     On June 2, 1998, Pennzoil issued 1,500,000 shares of cumulative preferred
stock at $100 per share. Dividends on the 6.49% Series A Cumulative Preferred
Stock, par value $1.00 per share, are cumulative from the date of original issue
and will be payable quarterly, in cash, when declared by the Board of Directors
of the Company, commencing September 30, 1998. Preferred dividends are required
to be deducted from net income in determining net income per common share. The
Series A Cumulative Preferred Stock will be redeemable at the option of Pennzoil
at any time on or after June 2, 2008, in whole or in part, at a redemption price
of $100 per share, plus accrued and unpaid dividends to the redemption date.


Item 2. Management's Discussion and Analysis of Financial Condition
        and Results of Operations

Results of Operations

     Net income available to common shareholders for the quarter and six months
ended June 30, 1998 was $8.1 million, or $0.17 per share, and $17.7 million, or
$0.37 per share, respectively. This compares with net income available to common
shareholders of $23.9 million, or $0.51 per share, for the second quarter of
1997 and $81.4 million, or $1.74 per share, for the six months ended June 30,
1997. The decrease in net income for the quarter and six months ended June 30,
1998 compared to the same periods in 1997 was primarily due to lower realized
liquids prices and lower natural gas production volumes partially offset by
higher results in the Motor Oil and Refined Products segment.

<PAGE>   10

           PART I. FINANCIAL INFORMATION - continued

Oil and Gas

     Operating income from this segment for the quarter and six months ended
June 30, 1998 was $24.9 million and $55.7 million, respectively. This compares
with operating income of $69.1 million and $176.3 million, respectively, for the
same periods in 1997. The decrease in operating income for the quarter ended
June 30, 1998, compared to the same period in 1997, was primarily due to lower
realized liquids prices, lower natural gas and liquids production volumes and
higher exploration expense partially offset by higher realized natural gas
prices and lower operating expense. The decrease in operating income for the six
months ended June 30, 1998, compared to the same period in 1997, was primarily
due to lower realized prices for natural gas and liquids, lower natural gas and
liquids production volumes and higher exploration expense.
     Natural gas price realizations averaged $2.12 per thousand cubic feet
("Mcf") and $2.09 per Mcf, respectively, for the quarter and six months ended
June 30, 1998, compared to $1.98 per Mcf and $2.34 per Mcf, respectively, for
the same periods in 1997. Liquids price realizations averaged $11.00 per barrel
and $11.70 per barrel for the quarter and six months ended June 30, 1998,
compared to $15.66 per barrel and $17.40 per barrel, respectively, for the same
periods in 1997. Natural gas volumes produced for sale were 481.8 million cubic
feet ("MMcf") per day and 490.6 MMcf per day, respectively, for the quarter and
six months ended June 30, 1998, compared to 616.4 MMcf per day and 570.1 MMcf
per day, respectively, for the same periods in 1997. Liquids production volumes
were 55.3 thousand barrels ("Mbbls") per day and 54.5 Mbbls per day,
respectively, for the quarter and six months ended June 30, 1998, compared to
62.8 Mbbls per day and 58.2 Mbbls per day, respectively, for the same periods in
1997.
     The decrease in production volumes for the quarter ended June 30, 1998
compared to the same period in 1997 is primarily due to the sale of Pennzoil's
Canadian natural gas assets in December 1997 and production decline at
Pennzoil's West Cameron 580 block in the Gulf of Mexico.
     In the Gulf of Mexico, a platform at Ship Shoal 150 came online in June
1998 and new wells at West Cameron 575 and West Cameron 291 will come on stream
during the remainder of 1998. Pennzoil is also continuing its in-fill drilling
program in the Carthage field in east Texas and its enhanced oil recovery
project in SACROC in west Texas. Pennzoil's domestic drilling program for the
second half of 1998 includes three South Louisiana exploration wells located in
Tiger Pass, Patterson and Red Fish Point fields and four south Texas exploration
wells in the Jennings, Escobas and Tom O'Connor fields. Pennzoil also plans to
spud additional offshore exploration wells before the end of 1998.
     Internationally, Pennzoil has drilled two of the seven planned exploration
wells for Australia, Azerbaijan and Egypt in 1998. Azerbaijan International
Operating Company, in which Pennzoil has a 4.8 percent carried interest,
delivered its first oil shipment to the Black Sea in March 1998 from the
Azeri-Chirag-Gunashli ("ACG") joint development area. The ACG joint development
area is estimated to contain nearly 5 billion barrels of crude oil. At the end
of the second quarter of 1998, daily crude oil production was 76,000 barrels.
Total daily production is expected to increase to 100,000 barrels by year-end
1998.

<PAGE>   11

           PART I. FINANCIAL INFORMATION - continued

     On the nearby Karabakh prospect, Caspian International Petroleum Company
("CIPCO"), the joint operating company in which Pennzoil has a 30 percent
interest, encountered natural gas in its first well drilled last year. CIPCO
began drilling a second well in April 1998 and encountered and tested two gas
zones with combined flow rates of 50 MMcf per day. A third well is scheduled for
the fourth quarter of 1998.
     In Egypt, Pennzoil has five concessions covering 9.2 million acres. Four
blocks are located in the Gulf of Suez: North July (100 percent Pennzoil), West
Feiran (50 percent Pennzoil), Southwest Gebel el-Zeit (87.5 percent Pennzoil)
and Southeast Gulf of Suez (50 percent Pennzoil). The fifth block, West Beni
Suef (100 percent Pennzoil), is located in Egypt's western desert. In June 1998,
Pennzoil announced an oil discovery on the North July block. The NJ-1 well,
Pennzoil's first well on the North July block, tested at a rate of 6,000 barrels
per day. In July 1998, Pennzoil began drilling an exploration well, south
Ashrafi-1B, on the Southwest Gebel el-Zeit block. Two additional exploratory
wells are planned for Egypt in 1998, including another at Southwest Gebel
el-Zeit and one at West Feiran. Pennzoil is also conducting a seismic program on
the West Beni Suef field that will continue for the rest of the year.
     On July 1, 1998, Pennzoil began operating the B2X-68/79 (54 percent
Pennzoil) and B2X-70/80 (45 percent Pennzoil) oil production blocks in
Venezuela. The two blocks are located on approximately 10,000 acres each in
eastern Lake Maracaibo. Pennzoil's net daily production from these two blocks is
approximately 2,500 barrels per day. Pennzoil also operates the 1.3 million acre
East Falcon block onshore in northwestern Venezuela and will commence a three
well development program in September 1998. Net production on East Falcon is
approximately 1,000 barrels per day.

Motor Oil & Refined Products

     Operating income from this segment for the quarter and six months ended
June 30, 1998 was $32.3 million and $57.2 million, respectively. This compares
to operating income of $25.8 million and $38.9 million, respectively, for the
same periods in 1997. The increase in operating income for the quarter and six
months ended June 30, 1998 was primarily due to increased motor oil volumes and
improved results from the Shreveport refinery upgrade, which was not fully
operational until the second quarter of 1997. The six months results were also
affected by improved results for the Excel Paralubes venture that operated below
capacity during the first quarter of 1997.
     Total domestic lubricating products volumes were up 4% in the second
quarter of 1998 over the same period in 1997 and margins remained strong.
Pennzoil motor oil continues to be the nation's top selling motor oil with a
U.S. market share in excess of 21%.

<PAGE>   12
           PART I. FINANCIAL INFORMATION - continued

Fast Lube Operations

     The fast lube operations segment, which operates through Jiffy Lube
International, Inc. ("Jiffy Lube"), recorded operating income of $6.9 million
and $11.6 million, respectively, for the quarter and six months ended June 30,
1998. This compares with operating income of $6.3 million and $10.8 million,
respectively, for the same periods in 1997. The increase in operating income for
the quarter and six months ended June 30, 1998 was primarily due to improved
company-operated store results and increased royalty income.
     Systemwide sales increased $13.7 million, or 7%, for the quarter ended June
30, 1998 to $208.6 million and $26.8 million, or 7.2%, for the six months ended
June 30, 1998 from comparable periods in 1997. Systemwide average ticket prices
for the quarter ended June 30, 1998 increased $1.08 to $37.22 and for the six
months ended June 30, 1998 increased $1.05 to $36.88, from comparable periods in
1997. There were 1,556 domestic lube centers (including 596 Jiffy Lube
company-operated centers) open as of June 30, 1998.
     In 1998, Jiffy Lube has opened 40 centers and plans to open an additional
80 by year-end 1998. As of June 30, 1998, there were 166 fast-oil change units
open in Sears Centers of which 135 are company operated.

Other

     Other operating income (loss) for the quarter and six months ended June 30,
1998 was $8.9 million and $19.3 million, respectively, compared with ($3.2)
million and $12.5 million for the same periods in 1997. Pennzoil's other income
includes dividend income of $10.9 million and $21.8 million for the quarter and
six months ended June 30, 1998, respectively, from its investment in common
stock of Chevron. Pennzoil beneficially owned approximately 18 million shares of
common stock of Chevron on June 30, 1998.
     Net interest expense for the quarter and six months ended June 30, 1998
increased $2.3 million and $7.1 million, respectively, from the same periods in
1997 primarily due to higher short-term borrowings and lower capitalized
interest.

Capital Resources and Liquidity

     Preferred Stock. On June 2, 1998, Pennzoil issued 1,500,000 shares of
cumulative preferred stock at $100 per share. Dividends on the 6.49% Series A
Cumulative Preferred Stock, par value $1.00 per share, are cumulative from the
date of original issue and will be payable quarterly, in cash, when declared by
the Board of Directors of the Company, commencing September 30, 1998. The Series
A Cumulative Preferred Stock will be redeemable at the option of Pennzoil at any
time on or after June 2, 2008, in whole or in part, at a redemption price of
$100 per share, plus accrued and unpaid dividends to the redemption date.

<PAGE>   13

           PART I. FINANCIAL INFORMATION - continued

     Cash Flow. As of June 30, 1998, Pennzoil had cash and cash equivalents of
$21.0 million. During the six months ended June 30, 1998, cash and cash
equivalents increased $2.4 million. Cash flows from operating activities for the
six months ended June 30, 1998 totaled $68.9 million.

     Debt Instruments and Repayments. During the six months ended June 30, 1998,
certain holders of Pennzoil's exchangeable debentures exchanged debentures for
Chevron common stock, in accordance with the respective supplemental indentures,
which resulted in a decrease of $.4 million of outstanding exchangeable
debentures. Reference is made to Note 4 of the Notes to Condensed Consolidated
Financial Statements for additional information on Pennzoil's exchangeable
debentures.
     Borrowings under Pennzoil's commercial paper, revolving credit facility and
other variable-rate credit arrangements totaled $509.8 million as of June 30,
1998, all of which has been classified as long-term debt.

Year 2000

     Pennzoil has begun the process of identifying, evaluating and implementing
new operating computer systems necessary to address potential year 2000
compliance issues. Many of Pennzoil's operating and financial systems are
already compliant. Pennzoil's remaining operating and financial systems are
scheduled for compliance in phases and will be compliant by the year 2000.
Pennzoil is communicating with software vendors, business partners and others
with which it conducts business to provide written assurances that their systems
will be year 2000 compliant. The total future cost associated with potential
year 2000 compliance issues has not been determined, but is not expected to have
a material adverse effect on the consolidated financial position or results of
operations of Pennzoil.

Forward-Looking Statements - Safe Harbor Provisions

     This quarterly report on Form 10-Q of Pennzoil Company for the quarter
ended June 30, 1998 contains certain forward-looking statements within the
meaning of Section 27A of the Securities Act of 1933, as amended, and Section
21E of the Securities Exchange Act of 1934, as amended, which are intended to be
covered by the safe harbors created thereby. To the extent that such statements
are not recitations of historical fact, such statements constitute
forward-looking statements which, by definition, involve risks and
uncertainties. Where, in any forward-looking statements, Pennzoil expresses an
expectation or belief as to future results or events, such expectation or belief
is expressed in good faith and believed to have a reasonable basis, but there
can be no assurance that the statement of expectation or belief will result or
be achieved or accomplished.
     The following are factors the could cause actual results or events to
differ materially from those anticipated, and include but are not limited to:
general economic, financial and business conditions; commodity prices for
natural gas and crude oil; the effect of weather on natural gas demand and
consumption; competition for international drilling rights; the costs of
exploration and development of petroleum reserves; exploration risks; political
risks impacting exploration and development; competition in the motor oil and
marketing business; base oil margins and supply and demand in the base oil
business; the success and cost of advertising and promotional efforts;
mechanical failure in refining operations; unanticipated environmental
liabilities; changes in and compliance with governmental regulations; changes in
tax laws; and the cost and effects of legal proceedings.



<PAGE>   14
                   PART I. FINANCIAL INFORMATION - continued
                                  (UNAUDITED)

<TABLE>
The following tables show revenues and operating income by segment, other
components of income and operating data.

<CAPTION>
                                                                      Three Months Ended                 Six Months Ended
                                                                           June 30                           June 30
                                                                 ----------------------------      ----------------------------
                                                                    1998             1997             1998             1997
                                                                 -----------      -----------      -----------      -----------
                                                                             (Dollar amounts expressed in thousands)
<S>                                                              <C>              <C>              <C>              <C>
REVENUES
   Oil and Gas                                                   $  157,928       $  204,274       $  318,957       $  431,015
   Motor Oil & Refined Products                                     422,277          453,095          795,173          886,048
   Fast Lube Operations                                              87,136           83,647          167,846          158,797
   Other                                                              8,664           (2,460)          18,531            9,009
   Intersegment sales                                               (43,502)         (90,199)        (122,194)        (187,512)
                                                                 -----------      -----------      -----------      -----------
        Total revenues                                           $  632,503       $  648,357       $1,178,313       $1,297,357
                                                                 -----------      -----------      -----------      -----------


OPERATING INCOME
   Oil and Gas                                                   $   24,861       $   69,107       $   55,676       $  176,286
   Motor Oil & Refined Products                                      32,339           25,802           57,179           38,851
   Fast Lube Operations                                               6,912            6,316           11,562           10,812
   Other                                                              8,905           (3,181)          19,339           12,484
                                                                 -----------      -----------      -----------      -----------
        Total operating income                                       73,017           98,044          143,756          238,433

Corporate administrative expenses                                    17,608           19,905           34,509           32,987
Interest charges, net                                                43,499           41,161           85,379           78,280
                                                                 -----------      -----------      -----------      -----------
Income before income tax                                             11,910           36,978           23,868          127,166

Income tax provision                                                  3,083           13,101            5,383           45,738
                                                                 -----------      -----------      -----------      -----------


NET INCOME                                                       $    8,827       $   23,877       $   18,485       $   81,428
                                                                 ===========      ===========      ===========      ===========


RATIO OF EARNINGS TO FIXED CHARGES                                                                       1.20             2.19
                                                                                                   ===========      ===========
</TABLE>



<PAGE>   15

                   PART I. FINANCIAL INFORMATION - continued
                                  (UNAUDITED)

<TABLE>
<CAPTION>
                                                                    Three Months Ended                    Six Months Ended
                                                                          June 30                              June 30
                                                               ------------------------------      ------------------------------
                                                                   1998              1997              1998              1997
                                                               ------------      ------------      ------------      ------------
<S>                                                            <C>               <C>               <C>               <C>
OPERATING DATA
--------------

OIL AND GAS
  Net production
    Crude oil, condensate and natural
      gas liquids (barrels per day)                                55,329            62,781            54,507            58,232
    Natural gas produced for sale (Mcf per day)                   481,808           616,360           490,608           570,111

  Weighted average prices
    Crude oil, condensate and natural
      gas liquids (per barrel)                                 $    11.00        $    15.66        $    11.70        $    17.40
    Natural gas (per Mcf)                                      $     2.12        $     1.98        $     2.09        $     2.34


MOTOR OIL & REFINED PRODUCTS
  Sales (barrels per day)
    Gasoline and naphtha                                           23,259            20,696            23,215            19,120
    Distillates and gas oils                                       25,864            24,943            25,704            27,702
    Lubricating oil and other specialty products<F1>               30,593            31,378            27,602            28,662
    Residual fuel oils                                              1,617             1,114             1,630             2,307
                                                               -----------       -----------       -----------       -----------
          Total sales (barrels per day)                            81,333            78,131            78,151            77,791
                                                               ===========       ===========       ===========       ===========

  Raw materials processed
      (barrels per day)<F2>                                        70,586            68,604            70,838            65,728

  Refining capacity
      (barrels per day)<F2>                                        80,300            76,000            80,300            76,000

FAST LUBE OPERATIONS
  Domestic systemwide sales (in thousands)                     $  208,594        $  194,859        $  400,441        $  373,564
  Same center sales (in thousands)                             $  197,435        $  193,679        $  378,585        $  371,358
  Centers open (U.S.)                                               1,556             1,441             1,556             1,441


<FN>
<F1>Excludes sales from Pennzoil's Penreco ownership in 1998. Pennzoil recorded
       Penreco's sales through September 30, 1997.
<F2>Includes Pennzoil's 50% ownership in Excel Paralubes.
</FN>

</TABLE>



<PAGE>   16

                           PART II. OTHER INFORMATION

Item 6.  Exhibits and Reports on Form 8-K

(a)  Exhibits -

     12  Computation of Ratio of Earnings to Fixed Charges for the six months
         ended June 30, 1998 and 1997.

     27  Financial Data Schedule

(b)  Reports -

     Pennzoil filed the following Current Reports on Form 8-K with the
     Securities and Exchange Commission:

     Date of Report           Items Reported

     May 20, 1998             Pro Forma financial information
                              relating to the Agreement and Plan
                              of Merger dated as of April 14, 1998
                              among Pennzoil Company, Pennzoil
                              Products Company, Downstream Merger
                              Company and Quaker State Company.

     May 28, 1998             Information relating to
                              Pennzoil Company's issue of 6.49%
                              Cumulative Preferred Stock, Series A.




<PAGE>   17


                             SIGNATURE



          Pursuant to the requirements of the Securities Exchange Act of 1934,
the Registrant has duly caused this report to be signed on its behalf by the
undersigned thereunto duly authorized.




                                   PENNZOIL COMPANY
                                      Registrant




                                   S/N Michael J. Maratea
                                   Michael J. Maratea
                                   Vice President and Controller




August 13, 1998



<PAGE>   18


                        PENNZOIL COMPANY AND SUBSIDIARIES
                                 INDEX TO EXHIBITS


Exhibit No.
-----------


         12    Computation of Ratio of Earnings to Fixed Charges for the six
               months ended June 30, 1998 and 1997.

         27    Financial Data Schedule