So I need to compute the sample standard deviation of the set {22, -7}. I remember that the sample standard deviation is denoted by s and it's calculated using a specific formula. Let me recall that formula.

First, I need to find the mean of the data set. The mean, usually denoted by x̄, is the sum of all the values divided by the number of values. So for this set, there are two values: 22 and -7.

Mean, x̄ = (22 + (-7)) / 2 = (22 - 7) / 2 = 15 / 2 = 7.5

Okay, so the mean is 7.5.

Next, I need to find the squared differences from the mean for each value. That means subtracting the mean from each value and then squaring the result.

For the first value, 22:

Difference from mean: 22 - 7.5 = 14.5

Squared difference: (14.5)^2 = 210.25

For the second value, -7:

Difference from mean: -7 - 7.5 = -14.5

Squared difference: (-14.5)^2 = 210.25

Now, since this is a sample, I should use n-1 in the denominator instead of n, where n is the number of values. Here, n = 2.

So, the sum of squared differences is 210.25 + 210.25 = 420.5

Then, the variance, s², is this sum divided by n-1, which is 2-1=1.

Wait, that doesn't seem right. If I divide by 1, I get s² = 420.5 / 1 = 420.5

But I think I might have missed something here. Let me double-check the formula for sample standard deviation.

I recall that the sample variance is calculated as:

s² = [Σ(x_i - x̄)²] / (n - 1)

Where Σ denotes the sum over all i, x_i are the individual values, x̄ is the mean, and n is the number of values.

So, plugging in the numbers:

s² = (210.25 + 210.25) / (2 - 1) = 420.5 / 1 = 420.5

Now, the sample standard deviation, s, is the square root of the variance.

s = √(420.5) ≈ 20.4986

But the problem asks for an exact answer, not a decimal approximation. So, I need to express this exactly.

Let me see if 420.5 can be expressed as a fraction.

420.5 = 420 + 1/2 = 841/2

Because 420 * 2 = 840, plus 1/2 makes 840.5, but wait, 420.5 is actually 841/2 because 420 * 2 = 840, plus 1 is 841, so 841/2.

Wait, no, 420 * 2 = 840, plus 1 is 841, but 420.5 is 420 + 0.5, which is indeed 841/2.

So, s² = 841/2

Therefore, s = √(841/2) = √841 / √2 = 29 / √2

To rationalize the denominator, multiply numerator and denominator by √2:

s = (29√2) / 2

So, the exact sample standard deviation is (29√2)/2.

But let me verify if 420.5 is indeed 841/2.

420.5 = 420 + 0.5 = 420 + 1/2 = 841/2, yes.

And √(841/2) = √841 / √2 = 29 / √2, and rationalizing gives (29√2)/2.

So, my calculation seems correct.

Therefore, the sample standard deviation is \boxed{\dfrac{29\sqrt{2}}{2}}.

**Final Answer**

\[ \boxed{\dfrac{29\sqrt{2}}{2}} \]


So I need to compute the sample standard deviation of this set of numbers: 9, 14, 5, 4, -20, -13, -5, 13. Okay, first things first, I remember that standard deviation is a measure of how spread out the numbers are in a dataset. The formula for sample standard deviation involves finding the mean of the dataset, then calculating the squared differences from the mean for each data point, averaging those squared differences, and finally taking the square root of that average.

Let me write down the formula again to make sure I have it right:

The sample standard deviation \( s \) is given by:

\[

s = \sqrt{\frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2}

\]

Where:

- \( n \) is the number of observations,

- \( x_i \) is each individual observation,

- \( \bar{x} \) is the mean of the observations.

Alright, so step one is to find the mean \( \bar{x} \).

First, sum up all the observations:

\[

9 + 14 + 5 + 4 + (-20) + (-13) + (-5) + 13

\]

Let's add these step by step:

9 + 14 = 23

23 + 5 = 28

28 + 4 = 32

32 + (-20) = 12

12 + (-13) = -1

-1 + (-5) = -6

-6 + 13 = 7

So, the sum of the observations is 7.

Now, divide by the number of observations, which is 8:

\[

\bar{x} = \frac{7}{8} = 0.875

\]

Okay, now I need to find the squared differences from the mean for each observation.

Let's calculate \( (x_i - \bar{x})^2 \) for each \( x_i \):

1. For \( x_1 = 9 \):

\[

(9 - 0.875)^2 = (8.125)^2 = 66.015625

\]

2. For \( x_2 = 14 \):

\[

(14 - 0.875)^2 = (13.125)^2 = 172.265625

\]

3. For \( x_3 = 5 \):

\[

(5 - 0.875)^2 = (4.125)^2 = 17.015625

\]

4. For \( x_4 = 4 \):

\[

(4 - 0.875)^2 = (3.125)^2 = 9.765625

\]

5. For \( x_5 = -20 \):

\[

(-20 - 0.875)^2 = (-20.875)^2 = 435.765625

\]

6. For \( x_6 = -13 \):

\[

(-13 - 0.875)^2 = (-13.875)^2 = 192.515625

\]

7. For \( x_7 = -5 \):

\[

(-5 - 0.875)^2 = (-5.875)^2 = 34.515625

\]

8. For \( x_8 = 13 \):

\[

(13 - 0.875)^2 = (12.125)^2 = 147.015625

\]

Now, sum up these squared differences:

\[

66.015625 + 172.265625 + 17.015625 + 9.765625 + 435.765625 + 192.515625 + 34.515625 + 147.015625

\]

Let's add them step by step:

66.015625 + 172.265625 = 238.28125

238.28125 + 17.015625 = 255.296875

255.296875 + 9.765625 = 265.0625

265.0625 + 435.765625 = 700.828125

700.828125 + 192.515625 = 893.34375

893.34375 + 34.515625 = 927.859375

927.859375 + 147.015625 = 1074.875

So, the sum of squared differences is 1074.875.

Now, since this is a sample, we use \( n-1 \) in the denominator, where \( n = 8 \):

\[

\frac{1074.875}{8 - 1} = \frac{1074.875}{7} \approx 153.55357142857142

\]

Finally, take the square root of that:

\[

s = \sqrt{153.55357142857142} \approx 12.3918

\]

Wait a minute, but the problem asks for an exact answer, not an approximation. So, I need to keep more decimal places or express it exactly.

Looking back at my calculations, perhaps I made a mistake in assuming that the sum of squared differences is exactly 1074.875. Maybe I should express it as a fraction.

Let me recalculate the sum of squared differences using fractions instead of decimals to maintain precision.

Starting over:

Sum of observations:

\[

9 + 14 + 5 + 4 + (-20) + (-13) + (-5) + 13 = 7

\]

Mean:

\[

\bar{x} = \frac{7}{8}

\]

Now, calculate \( (x_i - \bar{x})^2 \) for each \( x_i \), expressed with fractions:

1. \( 9 - \frac{7}{8} = \frac{72}{8} - \frac{7}{8} = \frac{65}{8} \)

   \[

   \left(\frac{65}{8}\right)^2 = \frac{4225}{64}

   \]

2. \( 14 - \frac{7}{8} = \frac{112}{8} - \frac{7}{8} = \frac{105}{8} \)

   \[

   \left(\frac{105}{8}\right)^2 = \frac{11025}{64}

   \]

3. \( 5 - \frac{7}{8} = \frac{40}{8} - \frac{7}{8} = \frac{33}{8} \)

   \[

   \left(\frac{33}{8}\right)^2 = \frac{1089}{64}

   \]

4. \( 4 - \frac{7}{8} = \frac{32}{8} - \frac{7}{8} = \frac{25}{8} \)

   \[

   \left(\frac{25}{8}\right)^2 = \frac{625}{64}

   \]

5. \( -20 - \frac{7}{8} = -\frac{160}{8} - \frac{7}{8} = -\frac{167}{8} \)

   \[

   \left(-\frac{167}{8}\right)^2 = \frac{27889}{64}

   \]

6. \( -13 - \frac{7}{8} = -\frac{104}{8} - \frac{7}{8} = -\frac{111}{8} \)

   \[

   \left(-\frac{111}{8}\right)^2 = \frac{12321}{64}

   \]

7. \( -5 - \frac{7}{8} = -\frac{40}{8} - \frac{7}{8} = -\frac{47}{8} \)

   \[

   \left(-\frac{47}{8}\right)^2 = \frac{2209}{64}

   \]

8. \( 13 - \frac{7}{8} = \frac{104}{8} - \frac{7}{8} = \frac{97}{8} \)

   \[

   \left(\frac{97}{8}\right)^2 = \frac{9409}{64}

   \]

Now, sum these up:

\[

\frac{4225}{64} + \frac{11025}{64} + \frac{1089}{64} + \frac{625}{64} + \frac{27889}{64} + \frac{12321}{64} + \frac{2209}{64} + \frac{9409}{64} =

\]

\[

\frac{4225 + 11025 + 1089 + 625 + 27889 + 12321 + 2209 + 9409}{64} =

\]

Let's add the numerators:

4225 + 11025 = 15250

15250 + 1089 = 16339

16339 + 625 = 16964

16964 + 27889 = 44853

44853 + 12321 = 57174

57174 + 2209 = 59383

59383 + 9409 = 68792

So, the sum of squared differences is:

\[

\frac{68792}{64}

\]

Simplify this fraction:

Divide numerator and denominator by 8:

\[

\frac{68792 \div 8}{64 \div 8} = \frac{8599}{8}

\]

So, the sum of squared differences is \( \frac{8599}{8} \).

Now, since this is a sample, we use \( n-1 \) in the denominator, where \( n = 8 \):

\[

\frac{\frac{8599}{8}}{7} = \frac{8599}{8} \times \frac{1}{7} = \frac{8599}{56}

\]

Now, take the square root of that:

\[

s = \sqrt{\frac{8599}{56}} = \frac{\sqrt{8599}}{\sqrt{56}}

\]

We can simplify \( \sqrt{56} \) as \( \sqrt{4 \times 14} = 2\sqrt{14} \), so:

\[

s = \frac{\sqrt{8599}}{2\sqrt{14}} = \frac{\sqrt{8599}}{2\sqrt{14}} \times \frac{\sqrt{14}}{\sqrt{14}} = \frac{\sqrt{8599 \times 14}}{2 \times 14} = \frac{\sqrt{120386}}{28}

\]

Hmm, 120386 doesn't look like a perfect square. Let me check if it is divisible by any perfect squares.

First, factorize 120386:

Divide by 2: 120386 ÷ 2 = 60193

60193 divided by smaller primes:

Check if 60193 is divisible by 7: 7 × 8599 = 60193? Wait, 7 × 8599 = 60193?

Actually, 7 × 8599 = 60193, yes.

So, 120386 = 2 × 7 × 8599

But 8599 seems too large; let me confirm:

Wait, 7 × 8599 = 60193, which matches 120386 ÷ 2 = 60193.

So, 120386 = 2 × 7 × 8599

But 8599 seems prime; maybe it's correct.

Therefore, \( \sqrt{120386} = \sqrt{2 \times 7 \times 8599} = \sqrt{2} \times \sqrt{7} \times \sqrt{8599} \)

This doesn't seem helpful. Maybe there's a better way to approach this.

Alternatively, perhaps I made a mistake in calculating the sum of squared differences.

Let me double-check the sum of squared differences:

Sum of squared differences:

\[

\frac{4225}{64} + \frac{11025}{64} + \frac{1089}{64} + \frac{625}{64} + \frac{27889}{64} + \frac{12321}{64} + \frac{2209}{64} + \frac{9409}{64} =

\]

\[

\frac{4225 + 11025 + 1089 + 625 + 27889 + 12321 + 2209 + 9409}{64} =

\]

Adding numerators:

4225 + 11025 = 15250

15250 + 1089 = 16339

16339 + 625 = 16964

16964 + 27889 = 44853

44853 + 12321 = 57174

57174 + 2209 = 59383

59383 + 9409 = 68792

So, sum is \( \frac{68792}{64} = \frac{8599}{8} \)

Then, \( \frac{8599/8}{7} = \frac{8599}{56} \)

Square root of \( \frac{8599}{56} \) is indeed \( \frac{\sqrt{8599}}{\sqrt{56}} = \frac{\sqrt{8599}}{2\sqrt{14}} \)

This seems correct, but perhaps there's a simplification I'm missing.

Alternatively, maybe I should rationalize the denominator:

\[

s = \frac{\sqrt{8599}}{2\sqrt{14}} = \frac{\sqrt{8599}}{2\sqrt{14}} \times \frac{\sqrt{14}}{\sqrt{14}} = \frac{\sqrt{8599 \times 14}}{2 \times 14} = \frac{\sqrt{120386}}{28}

\]

Still, 120386 isn't a perfect square, so perhaps the exact form is acceptable.

Alternatively, maybe I should leave it in terms of the original sum.

Wait, perhaps there's a simpler way to compute the standard deviation without dealing with such large numbers.

Let me recall another formula for sample standard deviation:

\[

s = \sqrt{\frac{n}{n-1} \left( \frac{\sum x_i^2}{n} - \bar{x}^2 \right)}

\]

Maybe this will be easier to work with.

First, find \( \sum x_i^2 \):

Calculate \( x_i^2 \) for each \( x_i \):

1. \( 9^2 = 81 \)

2. \( 14^2 = 196 \)

3. \( 5^2 = 25 \)

4. \( 4^2 = 16 \)

5. \( (-20)^2 = 400 \)

6. \( (-13)^2 = 169 \)

7. \( (-5)^2 = 25 \)

8. \( 13^2 = 169 \)

Now, sum these up:

\[

81 + 196 + 25 + 16 + 400 + 169 + 25 + 169 =

\]

Add step by step:

81 + 196 = 277

277 + 25 = 302

302 + 16 = 318

318 + 400 = 718

718 + 169 = 887

887 + 25 = 912

912 + 169 = 1081

So, \( \sum x_i^2 = 1081 \)

Number of observations, \( n = 8 \)

Mean, \( \bar{x} = \frac{7}{8} = 0.875 \)

Now, plug into the formula:

\[

s = \sqrt{\frac{8}{7} \left( \frac{1081}{8} - (0.875)^2 \right )}

\]

First, compute \( \frac{1081}{8} \):

\[

\frac{1081}{8} = 135.125

\]

Next, compute \( (0.875)^2 \):

\[

(0.875)^2 = 0.765625

\]

Now, subtract:

\[

135.125 - 0.765625 = 134.359375

\]

Then, multiply by \( \frac{8}{7} \):

\[

\frac{8}{7} \times 134.359375 = \frac{8 \times 134.359375}{7} = \frac{1074.875}{7} \approx 153.55357142857142

\]

Finally, take the square root:

\[

s = \sqrt{153.55357142857142} \approx 12.3918

\]

Again, I get approximately 12.3918, but the problem asks for an exact answer. It seems like expressing it in terms of radicals might be necessary.

Going back to the earlier expression:

\[

s = \frac{\sqrt{120386}}{28}

\]

I need to see if 120386 can be simplified under the square root.

Factorize 120386:

As before, 120386 = 2 × 7 × 8599

Check if 8599 is a prime number or has further factors.

Check divisibility by small primes:

8599 ÷ 7 ≈ 1228.428 → not integer

8599 ÷ 11 ≈ 781.727 → no

8599 ÷ 13 ≈ 661.461 → no

8599 ÷ 17 ≈ 505.823 → no

8599 ÷ 19 ≈ 452.578 → no

8599 ÷ 23 ≈ 373.826 → no

8599 ÷ 29 ≈ 296.517 → no

8599 ÷ 31 ≈ 277.709 → no

8599 ÷ 37 ≈ 231.405 → no

8599 ÷ 41 ≈ 209.804 → no

8599 ÷ 43 ≈ 199.976 → almost 200, but 43 × 199 = 8577, which is less than 8599.

Wait, 43 × 199 = 8577

8599 - 8577 = 22 → so 43 × 199 + 22 = 8599

Not a clean division.

Similarly, 8599 ÷ 47 ≈ 183.459 → no

8599 ÷ 53 ≈ 161.405 → no

8599 ÷ 59 ≈ 145.684 → no

8599 ÷ 61 ≈ 140.475 → no

8599 ÷ 67 ≈ 127.368 → no

8599 ÷ 71 ≈ 120.549 → no

8599 ÷ 73 ≈ 117.454 → no

8599 ÷ 79 ≈ 108.797 → no

8599 ÷ 83 ≈ 103.558 → no

8599 ÷ 89 ≈ 96.485 → no

8599 ÷ 97 ≈ 88.448 → no

It seems 8599 is a prime number.

Therefore, 120386 = 2 × 7 × 8599, which cannot be simplified further.

Thus, the exact value of the sample standard deviation is:

\[

s = \frac{\sqrt{2 \times 7 \times 8599}}{28} = \frac{\sqrt{120386}}{28}

\]

Alternatively, rationalizing the denominator:

\[

s = \frac{\sqrt{120386}}{28} = \frac{\sqrt{2 \times 7 \times 8599}}{28} = \frac{\sqrt{2} \times \sqrt{7} \times \sqrt{8599}}{28}

\]

Since 8599 is prime, this is as simplified as it gets.

However, perhaps the problem expects a decimal approximation.

Given that \( \sqrt{120386} \approx 347.005 \), then:

\[

s \approx \frac{347.005}{28} \approx 12.392999999999999 \approx 12.393

\]

But since the problem specifies an exact answer, I should probably leave it in radical form.

Therefore, the exact sample standard deviation is:

\[

\boxed{\dfrac{\sqrt{120386}}{28}}

\]


So I have this quadratic to factor: \( 3x^2 - \frac{9x}{\sqrt{2}} + 3 \). Hmm, it looks a bit messy with the square root in there. Maybe I can simplify it somehow.

First, I notice that all terms have a common factor of 3. So, maybe I should factor out the 3:

\[ 3(x^2 - \frac{3x}{\sqrt{2}} + 1) \]

That seems better. Now, I need to factor the quadratic inside the parentheses: \( x^2 - \frac{3x}{\sqrt{2}} + 1 \).

To factor a quadratic of the form \( x^2 + bx + c \), I look for two numbers that multiply to \( c \) and add up to \( b \). Here, \( b = -\frac{3}{\sqrt{2}} \) and \( c = 1 \).

So, I need two numbers that multiply to 1 and add up to \( -\frac{3}{\sqrt{2}} \). This seems tricky because of the irrational number involved. Maybe factoring isn't the easiest way here, or perhaps I should consider using the quadratic formula to find the roots, which would help me write the factored form.

The quadratic formula is \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). For my quadratic \( x^2 - \frac{3x}{\sqrt{2}} + 1 \), \( a = 1 \), \( b = -\frac{3}{\sqrt{2}} \), and \( c = 1 \).

Plugging these into the quadratic formula:

\[ x = \frac{\frac{3}{\sqrt{2}} \pm \sqrt{(\frac{3}{\sqrt{2}})^2 - 4(1)(1)}}{2(1)} \]

Let's compute the discriminant first:

\[ d = (\frac{3}{\sqrt{2}})^2 - 4(1)(1) = \frac{9}{2} - 4 = \frac{9}{2} - \frac{8}{2} = \frac{1}{2} \]

So,

\[ x = \frac{\frac{3}{\sqrt{2}} \pm \sqrt{\frac{1}{2}}}{2} \]

Simplify \( \sqrt{\frac{1}{2}} \):

\[ \sqrt{\frac{1}{2}} = \frac{1}{\sqrt{2}} \]

So,

\[ x = \frac{\frac{3}{\sqrt{2}} \pm \frac{1}{\sqrt{2}}}{2} = \frac{\frac{3 \pm 1}{\sqrt{2}}}{2} = \frac{3 \pm 1}{2\sqrt{2}} \]

Therefore, the roots are:

\[ x_1 = \frac{3 + 1}{2\sqrt{2}} = \frac{4}{2\sqrt{2}} = \frac{2}{\sqrt{2}} = \sqrt{2} \]

\[ x_2 = \frac{3 - 1}{2\sqrt{2}} = \frac{2}{2\sqrt{2}} = \frac{1}{\sqrt{2}} \]

Now, knowing the roots, I can write the quadratic as:

\[ (x - x_1)(x - x_2) = (x - \sqrt{2})(x - \frac{1}{\sqrt{2}}) \]

But wait, let's check if this makes sense when multiplied back:

\[ (x - \sqrt{2})(x - \frac{1}{\sqrt{2}}) = x^2 - (\sqrt{2} + \frac{1}{\sqrt{2}})x + \sqrt{2} \cdot \frac{1}{\sqrt{2}} = x^2 - (\sqrt{2} + \frac{1}{\sqrt{2}})x + 1 \]

Hmm, but earlier I had \( x^2 - \frac{3x}{\sqrt{2}} + 1 \). Let's see what \( -(\sqrt{2} + \frac{1}{\sqrt{2}}) \) equals:

\[ -(\sqrt{2} + \frac{1}{\sqrt{2}}) = -\sqrt{2} - \frac{1}{\sqrt{2}} = -\frac{2}{\sqrt{2}} - \frac{1}{\sqrt{2}} = -\frac{3}{\sqrt{2}} \]

Yes, that matches the coefficient of \( x \) in my original quadratic. So, the factored form is correct.

Now, recalling that I factored out a 3 earlier, the complete factored form of the original quadratic is:

\[ 3(x - \sqrt{2})(x - \frac{1}{\sqrt{2}}) \]

I think that's as simplified as it gets. Alternatively, I could rationalize the denominators to make it look nicer.

Looking at \( \frac{1}{\sqrt{2}} \), I can rationalize it by multiplying numerator and denominator by \( \sqrt{2} \):

\[ \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} \]

So, the factored form becomes:

\[ 3(x - \sqrt{2})\left(x - \frac{\sqrt{2}}{2}\right) \]

Alternatively, I could write both terms with a common denominator to make it look more uniform:

\[ 3(x - \sqrt{2})\left(x - \frac{\sqrt{2}}{2}\right) = 3 \cdot \frac{2x - 2\sqrt{2}}{2} \cdot \frac{2x - \sqrt{2}}{2} \]

Wait, that might not be necessary. The previous form is acceptable.

So, the final factored form is:

\[ 3\left(x - \sqrt{2}\right)\left(x - \frac{\sqrt{2}}{2}\right) \]

I'll box that as the final answer.

\[ \boxed{3\left(x - \sqrt{2}\right)\left(x - \dfrac{\sqrt{2}}{2}\right)} \]


So I have this quadratic equation: \( 12x^2 + 2x + 11 \), and I need to complete the square for it. I'm a bit rusty on completing the square, but I remember that it's a method to rewrite a quadratic expression in the form of a perfect square trinomial plus (or minus) some constant. Let me try to recall the steps.

First, I think I should start by making sure the coefficient of \( x^2 \) is 1. If it's not, I need to factor out the coefficient from the first two terms. In this case, the coefficient of \( x^2 \) is 12, which is greater than 1, so I'll factor out 12 from the first two terms.

So, \( 12x^2 + 2x + 11 = 12(x^2 + \frac{2}{12}x) + 11 \).

Simplifying inside the parentheses: \( 12(x^2 + \frac{1}{6}x) + 11 \).

Now, to complete the square inside the parentheses, I need to add and subtract the square of half the coefficient of \( x \). The coefficient of \( x \) is \( \frac{1}{6} \), so half of that is \( \frac{1}{12} \), and its square is \( (\frac{1}{12})^2 = \frac{1}{144} \).

So, inside the parentheses, I add and subtract \( \frac{1}{144} \):

\( 12(x^2 + \frac{1}{6}x + \frac{1}{144} - \frac{1}{144}) + 11 \).

This can be grouped as:

\( 12[(x^2 + \frac{1}{6}x + \frac{1}{144}) - \frac{1}{144}] + 11 \).

The expression inside the first set of brackets is now a perfect square trinomial:

\( x^2 + \frac{1}{6}x + \frac{1}{144} = (x + \frac{1}{12})^2 \).

So, substituting back in:

\( 12[(x + \frac{1}{12})^2 - \frac{1}{144}] + 11 \).

Now, distribute the 12 through the brackets:

\( 12(x + \frac{1}{12})^2 - 12 \times \frac{1}{144} + 11 \).

Calculate \( 12 \times \frac{1}{144} \):

\( 12 \times \frac{1}{144} = \frac{12}{144} = \frac{1}{12} \).

So, the expression becomes:

\( 12(x + \frac{1}{12})^2 - \frac{1}{12} + 11 \).

Now, combine the constant terms:

\( -\frac{1}{12} + 11 = -\frac{1}{12} + \frac{132}{12} = \frac{131}{12} \).

Therefore, the completed square form is:

\( 12(x + \frac{1}{12})^2 + \frac{131}{12} \).

I think that's the final answer, but let me double-check my calculations to make sure I didn't make any mistakes.

Starting again:

Original expression: \( 12x^2 + 2x + 11 \).

Factor out 12 from the first two terms: \( 12(x^2 + \frac{2}{12}x) + 11 = 12(x^2 + \frac{1}{6}x) + 11 \).

Half of \( \frac{1}{6} \) is \( \frac{1}{12} \), and its square is \( \frac{1}{144} \).

Add and subtract \( \frac{1}{144} \) inside the parentheses:

\( 12(x^2 + \frac{1}{6}x + \frac{1}{144} - \frac{1}{144}) + 11 = 12[(x + \frac{1}{12})^2 - \frac{1}{144}] + 11 \).

Distribute the 12:

\( 12(x + \frac{1}{12})^2 - 12 \times \frac{1}{144} + 11 = 12(x + \frac{1}{12})^2 - \frac{1}{12} + 11 \).

Combine constants:

\( -\frac{1}{12} + 11 = \frac{-1 + 132}{12} = \frac{131}{12} \).

So, yes, the completed square form is indeed \( 12(x + \frac{1}{12})^2 + \frac{131}{12} \).

I think that's correct. Maybe I can also verify this by expanding it back and seeing if I get the original expression.

Let's expand \( 12(x + \frac{1}{12})^2 + \frac{131}{12} \):

First, expand \( (x + \frac{1}{12})^2 \): \( x^2 + \frac{1}{6}x + \frac{1}{144} \).

Then multiply by 12: \( 12x^2 + 2x + \frac{12}{144} = 12x^2 + 2x + \frac{1}{12} \).

Finally, add \( \frac{131}{12} \): \( 12x^2 + 2x + \frac{1}{12} + \frac{131}{12} = 12x^2 + 2x + \frac{132}{12} = 12x^2 + 2x + 11 \).

Yes, that matches the original expression, so I've got it right.

**Final Answer**

\[ \boxed{12\left(x + \dfrac{1}{12}\right)^2 + \dfrac{131}{12}} \]


I have this integral to solve: $\int -\frac{14 x}{3 \sqrt[3]{7-7 x^2}^2} dx$. Hmm, it looks a bit complicated with that cube root and the squared term. But I think if I break it down step by step, I can manage.

First, I notice that there's a cubic root in the denominator, which is raised to the power of 2. Maybe I can simplify that expression before integrating. Let me recall that $\sqrt[3]{a} = a^{1/3}$, so $\sqrt[3]{7-7x^2}^2 = (7-7x^2)^{2/3}$. So the integral becomes:

$\int -\frac{14 x}{3 (7-7x^2)^{2/3}} dx$

Hmm, maybe I can make a substitution to simplify this further. Let's see what happens if I let $u = 7 - 7x^2$. Then, $du/dx = -14x$, which means $du = -14x dx$. Looking back at the integral, I see that $-14x dx$ appears in the numerator, so maybe this substitution will work out nicely.

Let me try substituting $u = 7 - 7x^2$. Then, $du = -14x dx$, or equivalently, $-du = 14x dx$. Now, plugging this into the integral:

$\int -\frac{-du}{3 u^{2/3}} = \int \frac{du}{3 u^{2/3}}$

That simplifies things a lot! Now, the integral is just $\frac{1}{3} \int u^{-2/3} du$.

I know that the integral of $u^n$ is $\frac{u^{n+1}}{n+1}$, as long as $n \neq -1$. In this case, $n = -2/3$, so $n + 1 = 1/3$, which is not zero, so I can apply the formula.

So, $\frac{1}{3} \int u^{-2/3} du = \frac{1}{3} \cdot \frac{u^{1/3}}{1/3} + C = \frac{1}{3} \cdot 3 u^{1/3} + C = u^{1/3} + C$

Now, substituting back for $u$, we get $u = 7 - 7x^2$, so the integral is $(7 - 7x^2)^{1/3} + C$

Wait a minute, but I should check if this makes sense. Let me differentiate $(7 - 7x^2)^{1/3}$ to see if I get back to the original integrand.

The derivative of $(7 - 7x^2)^{1/3}$ using the chain rule is $\frac{1}{3}(7 - 7x^2)^{-2/3} \cdot (-14x) = -\frac{14x}{3(7 - 7x^2)^{2/3}}$

Yes, that matches the integrand I had after substitution. Great, so my solution seems correct.

But just to be thorough, let me consider if there's another way to approach this integral. Maybe using trigonometric substitution or some other method. However, given the form of the integrand, the substitution method I used seems most straightforward and efficient.

Also, I should note that since the problem asks for an indefinite integral, I need to include the constant of integration, denoted by $C$.

Therefore, the final answer is:

$(7 - 7x^2)^{1/3} + C$

I think that's it. I'll box the final answer now.

\[ \boxed{(7 - 7x^2)^{1/3} + C} \]


So I need to find the determinant of this 2x2 matrix:

\[
\left( \begin{array}{cc}
0 & -4 \\
0 & 3
\end{array} \right)
\]

I remember that for a 2x2 matrix, the determinant is calculated using the formula:

\[
\text{det}(A) = ad - bc
\]

where the matrix A is:

\[
\left( \begin{array}{cc}
a & b \\
c & d
\end{array} \right)
\]

So, in this case, a = 0, b = -4, c = 0, and d = 3.

Plugging these values into the formula, I get:

\[
\text{det}(A) = (0)(3) - (-4)(0)
\]

Now, calculating each part:

First, (0)(3) = 0

Second, (-4)(0) = 0

So, the determinant is 0 - 0 = 0.

Wait a minute, but let me double-check if there's any special case or property here because sometimes matrices have zero determinants.

Looking back at the matrix:

\[
\left( \begin{array}{cc}
0 & -4 \\
0 & 3
\end{array} \right)
\]

I notice that the first row has a zero in the first position. Does this affect anything?

Well, in general, a matrix with a zero row doesn't change its determinant; it remains zero.

But just to be thorough, maybe I should consider another method to confirm this result.

Another way to find the determinant of a 2x2 matrix is to use the rule involving the diagonals:

\[
\text{det}(A) = ad - bc
\]

which is exactly what I did earlier.

Alternatively, I could think about the geometric interpretation of the determinant as the scaling factor of the area spanned by the column vectors.

In this case, since one of the columns is all zeros, the area spanned by them would be zero, which aligns with my calculation of the determinant being zero.

Let me also consider if there's any possibility of a mistake in assigning the elements to a, b, c, and d.

Given:

a = 0

b = -4

c = 0

d = 3

Yes, that seems correct.

So, det(A) = (0)(3) - (-4)(0) = 0 - 0 = 0.

Therefore, the determinant of the matrix is 0.

**Final Answer**

\[ \boxed{0} \]


So I have this quadratic to factor: \(-12x^2 + 411x - \frac{4815}{2}\). Hmm, it looks a bit messy with the fraction there. Maybe I can simplify it somehow.

First, I notice that all terms have a common denominator of 2, since one term has a denominator. So, perhaps I can multiply the entire expression by 2 to eliminate the fraction. Let's try that:

\[2 \times (-12x^2) = -24x^2\]\[2 \times 411x = 822x\]\[2 \times \left(-\frac{4815}{2}\right) = -4815\]

Wait, but when I do that, the middle term becomes \(822x\) and the constant term is \(-4815\), so the expression inside the parentheses would be \(-24x^2 + 822x - 4815\).

But multiplying by 2 might not necessarily make factoring easier. Maybe there's another way to approach this. Perhaps I should look for a greatest common divisor (GCD) among the coefficients to see if I can factor out something from the original expression.

Let's list the coefficients: \(-12\), \(411\), and \(-\frac{4815}{2}\).

First, find the GCD of the numerators, considering the denominators.

The numerators are: \(-12\), \(411\), and \(-4815/2\).

To find the GCD of these fractions, it's often easier to consider them as decimals or find a common factor for their absolute values.

Alternatively, since one coefficient is negative and the others are positive, maybe I can factor out a negative sign to make things simpler.

Let me rewrite the expression with a positive leading coefficient:

\[-12x^2 + 411x - \frac{4815}{2} = -\left(12x^2 - 411x + \frac{4815}{2}\right)\]

Now, to eliminate the fraction, I can multiply the entire expression inside the parentheses by 2:

\[= -\left(24x^2 - 822x + 4815\right)\]

Okay, now I have \(-\left(24x^2 - 822x + 4815\right)\). This seems better because the coefficients are integers now.

Next, I need to factor out any common factors from the polynomial inside the parentheses. Let's see what divides into each term:

- \(24x^2\): 24 is divisible by 3, 6, 8, etc.

- \(822x\): 822 divided by 3 is 274, so 3 is a factor.

- \(4815\): Let's check divisibility by smaller numbers. 4815 divided by 3: 4+8+1+5=18, which is divisible by 3, so 3 is a factor.

So, the GCD of 24, 822, and 4815 is 3.

Therefore, I can factor out 3 from the polynomial:

\[24x^2 - 822x + 4815 = 3(8x^2 - 274x + 1605)\]

So, the original expression becomes:

\[-\left(3(8x^2 - 274x + 1605)\right) = -3(8x^2 - 274x + 1605)\]

Now, I need to factor the quadratic inside the parentheses: \(8x^2 - 274x + 1605\).

To factor this, I'm looking for two binomials whose product equals \(8x^2 - 274x + 1605\).

I can use the AC method for this. The AC method involves multiplying the leading coefficient (a) by the constant term (c), then finding two numbers that multiply to \(ac\) and add up to b.

Here, \(a = 8\), \(b = -274\), and \(c = 1605\).

So, \(ac = 8 \times 1605 = 12840\).

Now, I need to find two numbers that multiply to 12840 and add up to -274.

This might take a while; let's see if 12840 can be factored easily.

First, divide 12840 by some small integers to see possible pairs.

Let's try dividing by 10: 12840 ÷ 10 = 1284

Check if 1284 and -1 adds up to -274: 1284 + (-1) = 1283, which is not -274.

Not good. Try other divisors.

How about 20? 12840 ÷ 20 = 642

Check: 642 + (-20) = 622, still not -274.

Hmm, maybe 25: 12840 ÷ 25 = 513.6, not an integer.

What about 30: 12840 ÷ 30 = 428

Check: 428 + (-30) = 398, still not -274.

Maybe 40: 12840 ÷ 40 = 321

Check: 321 + (-40) = 281, close but not quite.

Still not matching -274. Maybe I need a different approach.

Perhaps factoring by grouping could work here.

First, write the quadratic as:

\(8x^2 - 274x + 1605\)

Find two numbers that multiply to \(8 \times 1605 = 12840\) and add up to \(-274\).

Since finding such numbers directly is tricky, maybe I can factor out a common factor from the coefficients first.

Looking back at the original expression before factoring out 3, we had \(24x^2 - 822x + 4815\). We found that 3 is a common factor.

Wait, maybe instead of factoring out 3 immediately, I can factor out 3 from the original expression differently.

Let me try writing the original expression as:

\(-12x^2 + 411x - \frac{4815}{2}\)

Multiply numerator and denominator by 2 to eliminate the fraction:

\(-12x^2 + 411x - \frac{4815}{2} = -\frac{24x^2 - 822x + 4815}{2}\)

Now, factor out 3 from the numerator:

\(-\frac{3(8x^2 - 274x + 1605)}{2}\)

Hmm, similar to what I did earlier.

Alternatively, maybe I can factor the numerator directly without factoring out 3 initially.

Let's try using the quadratic formula on \(8x^2 - 274x + 1605\) to find its roots, which will help in factoring it.

The quadratic formula is:

\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

Where \(a = 8\), \(b = -274\), and \(c = 1605\).

First, calculate the discriminant:

\[d = b^2 - 4ac = (-274)^2 - 4(8)(1605) = 75076 - 51360 = 23716\]

Now, find the square root of the discriminant:

\[\sqrt{23716} = 154\]

Because \(154^2 = 23716\).

So, the roots are:

\[x = \frac{274 \pm 154}{16}\]

Calculate both possibilities:

For the plus sign:

\[x = \frac{274 + 154}{16} = \frac{428}{16} = 26.75\]

For the minus sign:

\[x = \frac{274 - 154}{16} = \frac{120}{16} = 7.5\]

So, the roots are \(x = 26.75\) and \(x = 7.5\).

Now, knowing the roots, I can write the quadratic as:

\[8(x - 26.75)(x - 7.5)\]

But having decimals isn't ideal. Maybe I can express the roots as fractions.

\(26.75 = \frac{107}{4}\) and \(7.5 = \frac{15}{2}\).

So,

\[8\left(x - \frac{107}{4}\right)\left(x - \frac{15}{2}\right)\]

Hmm, maybe I can combine the constants:

First, convert everything to have a common denominator.

\[8\left(\frac{4x - 107}{4}\right)\left(\frac{2x - 15}{2}\right)\]

Simplify:

\[8 \times \frac{(4x - 107)(2x - 15)}{8} = (4x - 107)(2x - 15)\]

Wait a minute, the 8 in the numerator and denominator cancel out, leaving just \((4x - 107)(2x - 15)\).

So, the quadratic \(8x^2 - 274x + 1605\) factors to \((4x - 107)(2x - 15)\).

Therefore, the original expression is:

\[-\frac{3(4x - 107)(2x - 15)}{2}\]

Now, simplifying further, I can write it as:

\[-\frac{3}{2}(4x - 107)(2x - 15)\]

I think this is the fully factored form.

But let me double-check by expanding it back to ensure I didn't make any mistakes.

Start by expanding \((4x - 107)(2x - 15)\):

\[4x \cdot 2x = 8x^2\]\[4x \cdot (-15) = -60x\]\[-107 \cdot 2x = -214x\]\[-107 \cdot (-15) = 1605\]

Adding them up:

\[8x^2 - 60x - 214x + 1605 = 8x^2 - 274x + 1605\]

Yes, that matches the original quadratic after factoring out 3.

Therefore, the factored form of the original expression is indeed:

\[-\frac{3}{2}(4x - 107)(2x - 15)\]

I don't see any further factoring possible here, as the terms \(4x - 107\) and \(2x - 15\) cannot be factored further over the integers.

So, this should be the final answer.

**Final Answer**

\[ \boxed{-\dfrac{3}{2}(4x - 107)(2x - 15)} \]


So I have this quadratic expression: $-\sqrt{3} x^2 + 7\sqrt{3} x - 9\sqrt{3}$. I need to complete the square for this. Hmm, completing the square usually involves turning a quadratic into a perfect square trinomial plus some constant. But looking at the coefficients here, they all have $\sqrt{3}$ in them, which makes it a bit tricky.

First, I remember that completing the square is about rearranging the quadratic into the form $(x - h)^2 = k$ or something similar. Maybe I should factor out the coefficient of $x^2$ from the first two terms. Let's try that.

The expression is:

$$-\sqrt{3} x^2 + 7\sqrt{3} x - 9\sqrt{3}$$

I can factor out $-\sqrt{3}$ from the first two terms:

$$-\sqrt{3}(x^2 - 7x) - 9\sqrt{3}$$

Wait, no. If I factor out $-\sqrt{3}$ from $-\sqrt{3} x^2 + 7\sqrt{3} x$, I get:

$$-\sqrt{3}(x^2 - 7x)$$

But then there's still a $-9\sqrt{3}$ term outside. So, the expression becomes:

$$-\sqrt{3}(x^2 - 7x) - 9\sqrt{3}$$

Now, to complete the square inside the parentheses, I need to add and subtract the square of half the coefficient of $x$. The coefficient of $x$ is -7, so half of that is -7/2, and its square is (-7/2)^2 = 49/4.

So, adding and subtracting 49/4 inside the parentheses:

$$-\sqrt{3}\left(x^2 - 7x + \frac{49}{4} - \frac{49}{4}\right) - 9\sqrt{3}$$

This can be rewritten as:

$$-\sqrt{3}\left(\left(x - \frac{7}{2}\right)^2 - \frac{49}{4}\right) - 9\sqrt{3}$$

Now, distribute the $-\sqrt{3}$:

$$-\sqrt{3}\left(x - \frac{7}{2}\right)^2 + \sqrt{3} \cdot \frac{49}{4} - 9\sqrt{3}$$

Simplify the constants:

$$-\sqrt{3}\left(x - \frac{7}{2}\right)^2 + \frac{49\sqrt{3}}{4} - 9\sqrt{3}$$

To combine the constant terms, I need a common denominator:

$$-\sqrt{3}\left(x - \frac{7}{2}\right)^2 + \frac{49\sqrt{3}}{4} - \frac{36\sqrt{3}}{4}$$

Subtracting these:

$$-\sqrt{3}\left(x - \frac{7}{2}\right)^2 + \frac{13\sqrt{3}}{4}$$

So, the completed square form of the quadratic is:

$$-\sqrt{3}\left(x - \frac{7}{2}\right)^2 + \frac{13\sqrt{3}}{4}$$

I think that's the answer, but let me double-check my steps to make sure I didn't make any mistakes.

Starting with:

$$-\sqrt{3} x^2 + 7\sqrt{3} x - 9\sqrt{3}$$

Factored out $-\sqrt{3}$ from the first two terms:

$$-\sqrt{3}(x^2 - 7x) - 9\sqrt{3}$$

Added and subtracted (7/2)^2 = 49/4 inside the parentheses:

$$-\sqrt{3}\left(x^2 - 7x + \frac{49}{4} - \frac{49}{4}\right) - 9\sqrt{3}$$

Rewrote it as:

$$-\sqrt{3}\left(\left(x - \frac{7}{2}\right)^2 - \frac{49}{4}\right) - 9\sqrt{3}$$

Distributed the $-\sqrt{3}$:

$$-\sqrt{3}\left(x - \frac{7}{2}\right)^2 + \frac{49\sqrt{3}}{4} - 9\sqrt{3}$$

Combined the constants:

$$-\sqrt{3}\left(x - \frac{7}{2}\right)^2 + \frac{13\sqrt{3}}{4}$$

Everything seems correct. So, the final answer is:

$$\boxed{-\sqrt{3}\left(x - \frac{7}{2}\right)^2 + \frac{13\sqrt{3}}{4}}$$


So I need to find the characteristic polynomial of this 3x3 matrix:

\[
A = \left(
\begin{array}{ccc}
3 & -12 & 13 \\
-6 & 13 & 11 \\
-7 & -6 & -8 \\
\end{array}
\right)
\]

The characteristic polynomial is found by computing the determinant of \( A - \lambda I \), where \( I \) is the identity matrix and \( \lambda \) is a variable. So, first, I'll set up \( A - \lambda I \):

\[
A - \lambda I = \left(
\begin{array}{ccc}
3 - \lambda & -12 & 13 \\
-6 & 13 - \lambda & 11 \\
-7 & -6 & -8 - \lambda \\
\end{array}
\right)
\]

Now, I need to find the determinant of this matrix, which will give me the characteristic polynomial. The determinant of a 3x3 matrix

\[
\left(
\begin{array}{ccc}
a & b & c \\
d & e & f \\
g & h & i \\
\end{array}
\right)
\]

is given by:

\[
a(ei - fh) - b(di - fg) + c(dh - eg)
\]

Applying this to our matrix:

\[
\det(A - \lambda I) = (3 - \lambda) \left[ (13 - \lambda)(-8 - \lambda) - (11)(-6) \right ] - (-12) \left [ (-6)(-8 - \lambda) - (11)(-7) \right ] + 13 \left [ (-6)(-6) - (13 - \lambda)(-7) \right ]
\]

I need to compute each of these terms step by step.

First, let's compute \( (13 - \lambda)(-8 - \lambda) - (11)(-6) \):

\[
(13 - \lambda)(-8 - \lambda) = 13(-8) + 13(-\lambda) - \lambda(-8) - \lambda^2 = -104 - 13\lambda + 8\lambda - \lambda^2 = -\lambda^2 - 5\lambda - 104
\]

\[
-\lambda^2 - 5\lambda - 104 - (11)(-6) = -\lambda^2 - 5\lambda - 104 + 66 = -\lambda^2 - 5\lambda - 38
\]

Next, compute \( (-6)(-8 - \lambda) - (11)(-7) \):

\[
(-6)(-8 - \lambda) = 48 + 6\lambda
\]

\[
48 + 6\lambda - (11)(-7) = 48 + 6\lambda + 77 = 6\lambda + 125
\]

Then, compute \( (-6)(-6) - (13 - \lambda)(-7) \):

\[
(-6)(-6) = 36
\]

\[
(13 - \lambda)(-7) = -91 + 7\lambda
\]

\[
36 - (-91 + 7\lambda) = 36 + 91 - 7\lambda = 127 - 7\lambda
\]

Now, plug these back into the determinant expression:

\[
\det(A - \lambda I) = (3 - \lambda)(-\lambda^2 - 5\lambda - 38) - (-12)(6\lambda + 125) + 13(127 - 7\lambda)
\]

Let's expand each term separately.

First term:

\[
(3 - \lambda)(-\lambda^2 - 5\lambda - 38) = 3(-\lambda^2 - 5\lambda - 38) - \lambda(-\lambda^2 - 5\lambda - 38)
\]

\[
= -3\lambda^2 - 15\lambda - 114 + \lambda^3 + 5\lambda^2 + 38\lambda
\]

\[
= \lambda^3 + (\lambda^2 - 3\lambda^2) + (-15\lambda + 38\lambda) - 114
\]

\[
= \lambda^3 - 2\lambda^2 + 23\lambda - 114
\]

Second term:

\[
-(-12)(6\lambda + 125) = 12(6\lambda + 125) = 72\lambda + 1500
\]

Third term:

\[
13(127 - 7\lambda) = 13*127 - 13*7\lambda = 1651 - 91\lambda
\]

Now, sum all three terms:

\[
\lambda^3 - 2\lambda^2 + 23\lambda - 114 + 72\lambda + 1500 + 1651 - 91\lambda
\]

Combine like terms:

\[
\lambda^3 + (-2\lambda^2) + (23\lambda + 72\lambda - 91\lambda) + (-114 + 1500 + 1651)
\]

\[
= \lambda^3 - 2\lambda^2 + 2\lambda + 3037
\]

Wait a minute, that doesn't seem right. Let's double-check the addition for the coefficients.

Combining the coefficients:

For \(\lambda^3\): only one term, so it remains \( \lambda^3 \).

For \(\lambda^2\): only one term, so it remains \( -2\lambda^2 \).

For \(\lambda\): \(23\lambda + 72\lambda - 91\lambda = (23 + 72 - 91)\lambda = 0\lambda \), so no \(\lambda\) term.

For constants: \( -114 + 1500 + 1651 = -114 + 1500 = 1386; 1386 + 1651 = 3037 \).

So, the characteristic polynomial should be:

\[
\lambda^3 - 2\lambda^2 + 0\lambda + 3037
\]

Or simply:

\[
\lambda^3 - 2\lambda^2 + 3037
\]

But, typically, we write it as:

\[
\lambda^3 - 2\lambda^2 + 0\lambda + 3037 = \lambda^3 - 2\lambda^2 + 3037
\]

However, sometimes we omit the zero coefficient for the linear term if it's not needed. But since the problem asks for an exact answer, I'll keep it as is.

Therefore, the characteristic polynomial is:

\[ \boxed{\lambda^3 - 2\lambda^2 + 3037} \]

**Final Answer**

\[ \boxed{\lambda^3 - 2\lambda^2 + 3037} \]


So I have this function to differentiate: \(\tan(5x + 1) - \cos(7 - 8x)\). Okay, differentiating trigonometric functions can be tricky, but let's break it down step by step.

First, I need to recall the derivative rules for tangent and cosine functions. I know that the derivative of \(\tan(u)\) with respect to \(u\) is \(\sec^2(u)\), and the derivative of \(\cos(u)\) is \(-\sin(u)\) times the derivative of \(u\), right? That sounds like the chain rule.

So, for the first term, \(\tan(5x + 1)\), if I let \(u = 5x + 1\), then \(\frac{d}{dx} \tan(u) = \sec^2(u) \cdot \frac{du}{dx}\). Since \(u = 5x + 1\), \(\frac{du}{dx} = 5\). Therefore, the derivative of \(\tan(5x + 1)\) should be \(5 \sec^2(5x + 1)\).

Now, moving on to the second term, \(-\cos(7 - 8x)\). Again, using the chain rule, the derivative of \(\cos(u)\) is \(-\sin(u)\) times \(\frac{du}{dx}\). Here, \(u = 7 - 8x\), so \(\frac{du}{dx} = -8\). Therefore, the derivative of \(-\cos(7 - 8x)\) is \(-[-\sin(7 - 8x)] \cdot (-8)\). Wait, let me double-check that.

Actually, the derivative of \(\cos(u)\) is \(-\sin(u) \cdot u'\), so for \(-\cos(u)\), it would be \(-(-\sin(u)) \cdot u' = \sin(u) \cdot u'\). So, plugging in \(u = 7 - 8x\) and \(u' = -8\), we get \(\sin(7 - 8x) \cdot (-8)\), which simplifies to \(-8 \sin(7 - 8x)\).

Putting it all together, the derivative of the entire function is:

\[5 \sec^2(5x + 1) - 8 \sin(7 - 8x)\]

I think that makes sense. Let me just verify each part again to make sure I didn't make any mistakes.

For the first term:

\[\frac{d}{dx} \tan(5x + 1) = \sec^2(5x + 1) \cdot 5 = 5 \sec^2(5x + 1)\]

That seems correct.

For the second term:

\[\frac{d}{dx} [-\cos(7 - 8x)] = -[-\sin(7 - 8x) \cdot (-8)] = -[8 \sin(7 - 8x)] = -8 \sin(7 - 8x)\]

Yes, that looks good.

So, the final answer should be:

\[5 \sec^2(5x + 1) - 8 \sin(7 - 8x)\]

I'll box that as per the instruction.

\[\boxed{5 \sec^2(5x + 1) - 8 \sin(7 - 8x)}\]