<PAGE>

                                                                    Exhibit 10.5


                                 July 12, 2000



Mr. C. E. "Gene" Miller
Eagle Investments Inc.
P.O. Box 348
Traverse City, MI   49685-0348

Dear Gene,

     After numerous discussions and detailed review, Miller Exploration
(hereinafter "MEXP") will agree to purchase the following interest from Eagle
Investments (hereinafter "Eagle"), subject to Board approval and (to the extent
                                                             ---
necessary) shareholder approval.  As you are aware, we both need to present a
purchase and sale agreement that would withstand the highest scrutiny, due to
the roles you play both in MEXP and Eagle.

     Pursuant to the letter dated April 1, 1999 (attached as Exhibit "A"), MEXP
had specific restrictions regarding the right to repurchase and certain rights
which have now expired.  Subject to the terms and conditions described below,
Eagle will contribute capital, extend said agreement, and modify certain rights
as detailed below.  It is therefore agreed:

     1)  Eagle will infuse five hundred thousand ($500,000) dollars into MEXP in
         exchange for common stock at $1.35/share.  Eagle will wire said funds
         to MEXP within five (5) business days of Board approval or shareholder
         approval if required.

     2)  MEXP will purchase an undivided 50% of Eagle's non-producing leasehold
         in the Mississippi Salt Basin pursuant to the interest represented in
         Exhibit "B" for a purchase price of two million ($2,000,000) dollars.
         Consideration will be paid in the form of MEXP common stock @
         $1.35/share.

     3)  MEXP's purchase of the above will be subject to shareholder approval of
         the Guardian Energy Management Corporation equity infusion of five
         million ($5,000,000) dollars.

     4)  Eagle will be granted warrants as follows:

<TABLE>
<CAPTION>

               Warrants    Exercise       Term
               --------    --------       ----
<S>                        <C>            <C>
               781,250      $1.35/share    1 year from the date of shareholder approval
               1,250,000    $2.50/share    2 years from the date of shareholder approval
</TABLE>
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Mr. C. E. "Gene" Miller
Page 2
July 12, 2000

     5)  Eagle will be responsible for its proportionate share of the working
         interest to drill the proposed wells on the salt domes pursuant to
         Exhibit "B".  Upon shareholder approval, MEXP will reimburse Eagle for
         50% of its cost to drill and/or complete the respective wells.\

     6)  MEXP's purchase will exclude any rights with respect to Eagle's
         interest in the existing wellbores, specifically the Allar 32-6 and
         Frost 5-11 #1 (Moselle Dome), Allar #6 (Midway Dome), Campbell #1
         (Centerville Dome), and Fairchild #1 (Richmond Dome).  In the event of
         a side track, mechanical difficulties or development location, Eagle
         will retain its rights in said proposed well(s).

     7)  MEXP will acquire all rights in the Langston #1 drilling unit (Kola
         Dome), excepting the Sligo Formation, which is the current producing
                ---------
         zone in the Langston #1.

       A)  Eagle will reserve the Sligo formation rights in the 640-acre
           drilling units immediately adjoining and abutting the Langston #1
           drilling unit. In the event a subsequent well is proposed to the
           Hosston formation, or a depth greater than the Sligo, Eagle will only
           have the rights of participation to the extent of its retained
           working interest in the deeper horizons. However, in the event that
           the subsequent well(s) is completed in the Hosston or deeper
           formation(s), and it is then determined within 120 days of reaching
           total depth, that the zones tested are considered to be unproductive
           or do not produce in paying quantities, and the well is subsequently
           completed in the Sligo formation, Eagle will participate, at its
           option, for its proportionate interest as if no sale had taken place
           to MEXP. Upon an election by Eagle to participate, Eagle will
           reimburse MEXP for its proportionate cost of the additional interest
           acquired as if the well had been drilled to the Sligo formation. If
           the above 120-day term expires and the well is subsequently completed
           in the Sligo formation, Eagle will participate as if no sale had
           taken place to MEXP for its rights in the Sligo formation completion
           without reimbursement for the drilling.

     8)    MEXP will have an exclusive right of first offer on any sales of
           Eagle's assets, including the remaining non-producing leasehold as
           well as the producing leasehold, in the counties which contain the
           existing MEXP leasehold base. MEXP will have a period of 15 days from
           receipt of such offer to reject or
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Mr. C. E. "Gene" Miller
Page 3
July 12, 2000


           accept the offer ("response period"). In the event of rejection (or
           no response by MEXP before the end of the response period), Eagle
           will be free and clear to sell or assign such interest to any third
           party at the terms and conditions equal to or greater than the terms
           offered to MEXP. The right of first offer will include a notice from
           Eagle to MEXP its desire to sell the producing and/or non-producing
           leasehold. At such time MEXP will have the right to present an offer
           within the response period described. Additionally, in the event
           Eagle receives an unsolicited offer from a third party, which is
           acceptable to Eagle, Eagle will also give MEXP the right of first
           refusal to accept the terms and conditions provided therein within
           the response period. The right of first offer will expire on
           December 31, 2000, unless extended by the mutual consent of both
           parties.

     9)    The purchase contemplated will be subject to the approval of the
           outside directors of MEXP. A copy of the Resolution approving said
           sale will be provided to Eagle.

     10)   This letter agreement contains the entire agreement between the
           parties with respect to the subject matter hereof, and there are no
           promises, terms, conditions, or obligations other than those
           contained herein.

     11)   This letter agreement and the rights and obligations of the party
           will be governed and interpreted in accordance with the laws of the
           State of Michigan without giving effect to principals of conflicts or
           laws.

     If the above meets with your approval, we would appreciate you signing in
the space provided below.  This offer must be accepted on or before July 15,
2000, or the proposal contained herein will be null and void.

                              Sincerely,

                              MILLER EXPLORATION COMPANY


                              Kelly E. Miller
                              President and CEO
KEM/sc
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Mr. C. E. "Gene" Miller
Page 4
July 12, 2000


      Agreed to and accepted this ______ day of ______________, 2000.

      EAGLE INVESTMENTS INC.


      By: _________________________________
           C. E. "Gene" Miller, President