Pre-Algebra Examples

$f (x) = \frac{8 x - 7}{7 x^{2} - 26 x - 45}$

Step 1

Write $f (x) = \frac{8 x - 7}{7 x^{2} - 26 x - 45}$ as an equation.

$y = \frac{8 x - 7}{7 x^{2} - 26 x - 45}$

Step 2

Simplify $\frac{8 x - 7}{7 x^{2} - 26 x - 45}$ .

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Step 2.1

Factor by grouping.

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Step 2.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 7 \cdot - 45 = - 315$ and whose sum is $b = - 26$ .

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Step 2.1.1.1

Factor $- 26$ out of $- 26 x$ .

$y = \frac{8 x - 7}{7 x^{2} - 26 (x) - 45}$

Step 2.1.1.2

Rewrite $- 26$ as $9$ plus $- 35$

$y = \frac{8 x - 7}{7 x^{2} + (9 - 35) x - 45}$

Step 2.1.1.3

Apply the distributive property.

$y = \frac{8 x - 7}{7 x^{2} + 9 x - 35 x - 45}$

Step 2.1.2

Factor out the greatest common factor from each group.

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Step 2.1.2.1

Group the first two terms and the last two terms.

$y = \frac{8 x - 7}{(7 x^{2} + 9 x) - 35 x - 45}$

Step 2.1.2.2

Factor out the greatest common factor (GCF) from each group.

$y = \frac{8 x - 7}{x (7 x + 9) - 5 (7 x + 9)}$

Step 2.1.3

Factor the polynomial by factoring out the greatest common factor, $7 x + 9$ .

$y = \frac{8 x - 7}{(7 x + 9) (x - 5)}$

Step 2.2

Split the fraction $\frac{8 x - 7}{(7 x + 9) (x - 5)}$ into two fractions.

$y = \frac{8 x}{(7 x + 9) (x - 5)} + \frac{- 7}{(7 x + 9) (x - 5)}$

Step 2.3

Move the negative in front of the fraction.

$y = \frac{8 x}{(7 x + 9) (x - 5)} - \frac{7}{(7 x + 9) (x - 5)}$

Step 3

The given equation $y = \frac{8 x}{(7 x + 9) (x - 5)} - \frac{7}{(7 x + 9) (x - 5)}$ can not be written as $y = k x^{2}$ , so $y$ doesn't vary directly with $x^{2}$ .

$y$ doesn't vary directly with $x$