Find the Quotient of the Two Functions f(x)=2x^2-x-6 g(x)=x^2-4

$f (x) = 2 x^{2} - x - 6$ $g (x) = x^{2} - 4$

Step 1

Replace the function designators with the actual functions in $\frac{f (x)}{g (x)}$ .

$\frac{2 x^{2} - x - 6}{x^{2} - 4}$

Step 2

Simplify.

Factor by grouping.

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 $a \cdot c = 2 \cdot - 6 = - 12$ and whose sum is $b = - 1$ .

Factor $- 1$ out of $- x$ .

$\frac{2 x^{2} - (x) - 6}{x^{2} - 4}$

Rewrite $- 1$ as $3$ plus $- 4$

$\frac{2 x^{2} + (3 - 4) x - 6}{x^{2} - 4}$

Apply the distributive property.

$\frac{2 x^{2} + 3 x - 4 x - 6}{x^{2} - 4}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(2 x^{2} + 3 x) - 4 x - 6}{x^{2} - 4}$

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

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

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

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

Simplify the denominator.

Rewrite $4$ as $2^{2}$ .

$\frac{(2 x + 3) (x - 2)}{x^{2} - 2^{2}}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 2$ .

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

Cancel the common factor of $x - 2$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{2 x + 3}{x + 2}$

Find the Quotient of the Two Functions f(x)=2x^2-x-6 g(x)=x^2-4

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