Pre-Algebra Examples

$f (x) = \frac{2 x^{2} + 5 x - 12}{x + 4}$

Step 1

Write $f (x) = \frac{2 x^{2} + 5 x - 12}{x + 4}$ as an equation.

$y = \frac{2 x^{2} + 5 x - 12}{x + 4}$

Step 2

Simplify $\frac{2 x^{2} + 5 x - 12}{x + 4}$ .

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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 $a \cdot c = 2 \cdot - 12 = - 24$ and whose sum is $b = 5$ .

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Step 2.1.1.1

Factor $5$ out of $5 x$ .

$y = \frac{2 x^{2} + 5 (x) - 12}{x + 4}$

Step 2.1.1.2

Rewrite $5$ as $- 3$ plus $8$

$y = \frac{2 x^{2} + (- 3 + 8) x - 12}{x + 4}$

Step 2.1.1.3

Apply the distributive property.

$y = \frac{2 x^{2} - 3 x + 8 x - 12}{x + 4}$

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.

$y = \frac{(2 x^{2} - 3 x) + 8 x - 12}{x + 4}$

Step 2.1.2.2

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

$y = \frac{x (2 x - 3) + 4 (2 x - 3)}{x + 4}$

Step 2.1.3

Factor the polynomial by factoring out the greatest common factor, $2 x - 3$ .

$y = \frac{(2 x - 3) (x + 4)}{x + 4}$

Step 2.2

Cancel the common factor of $x + 4$ .

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Step 2.2.1

Cancel the common factor.

$y = \frac{(2 x - 3) (x + 4)}{x + 4}$

Step 2.2.2

Divide $2 x - 3$ by $1$ .

$y = 2 x - 3$

Step 3

The given equation $y = 2 x - 3$ can not be written as $y = k x^{2}$ , so $y$ doesn't vary directly with $x^{2}$ .

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