Simplify (-12x^2+6x+90)/(6x^2-54)

$\frac{- 12 x^{2} + 6 x + 90}{6 x^{2} - 54}$

Step 1

Cancel the common factor of $- 12 x^{2} + 6 x + 90$ and $6 x^{2} - 54$ .

Factor $6$ out of $- 12 x^{2}$ .

$\frac{6 (- 2 x^{2}) + 6 x + 90}{6 x^{2} - 54}$

Factor $6$ out of $6 x$ .

$\frac{6 (- 2 x^{2}) + 6 (x) + 90}{6 x^{2} - 54}$

Factor $6$ out of $6 (- 2 x^{2}) + 6 (x)$ .

$\frac{6 (- 2 x^{2} + x) + 90}{6 x^{2} - 54}$

Factor $6$ out of $90$ .

$\frac{6 (- 2 x^{2} + x) + 6 \cdot 15}{6 x^{2} - 54}$

Factor $6$ out of $6 (- 2 x^{2} + x) + 6 (15)$ .

$\frac{6 (- 2 x^{2} + x + 15)}{6 x^{2} - 54}$

Cancel the common factors.

Factor $6$ out of $6 x^{2}$ .

$\frac{6 (- 2 x^{2} + x + 15)}{6 (x^{2}) - 54}$

Factor $6$ out of $- 54$ .

$\frac{6 (- 2 x^{2} + x + 15)}{6 x^{2} + 6 \cdot - 9}$

Factor $6$ out of $6 x^{2} + 6 \cdot - 9$ .

$\frac{6 (- 2 x^{2} + x + 15)}{6 (x^{2} - 9)}$

Cancel the common factor.

$\frac{6 (- 2 x^{2} + x + 15)}{6 (x^{2} - 9)}$

Rewrite the expression.

$\frac{- 2 x^{2} + x + 15}{x^{2} - 9}$

Step 2

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 $a \cdot c = - 2 \cdot 15 = - 30$ and whose sum is $b = 1$ .

Multiply by $1$ .

$\frac{- 2 x^{2} + 1 x + 15}{x^{2} - 9}$

Rewrite $1$ as $- 5$ plus $6$

$\frac{- 2 x^{2} + (- 5 + 6) x + 15}{x^{2} - 9}$

Apply the distributive property.

$\frac{- 2 x^{2} - 5 x + 6 x + 15}{x^{2} - 9}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(- 2 x^{2} - 5 x) + 6 x + 15}{x^{2} - 9}$

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

$\frac{x (- 2 x - 5) - 3 (- 2 x - 5)}{x^{2} - 9}$

Factor the polynomial by factoring out the greatest common factor, $- 2 x - 5$ .

$\frac{(- 2 x - 5) (x - 3)}{x^{2} - 9}$

Step 3

Simplify the denominator.

Rewrite $9$ as $3^{2}$ .

$\frac{(- 2 x - 5) (x - 3)}{x^{2} - 3^{2}}$

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 $b = 3$ .

$\frac{(- 2 x - 5) (x - 3)}{(x + 3) (x - 3)}$

Step 4

Simplify terms.

Cancel the common factor of $x - 3$ .

Cancel the common factor.

$\frac{(- 2 x - 5) (x - 3)}{(x + 3) (x - 3)}$

Rewrite the expression.

$\frac{- 2 x - 5}{x + 3}$

Factor $- 1$ out of $- 2 x$ .

$\frac{- (2 x) - 5}{x + 3}$

Rewrite $- 5$ as $- 1 (5)$ .

$\frac{- (2 x) - 1 (5)}{x + 3}$

Factor $- 1$ out of $- (2 x) - 1 (5)$ .

$\frac{- (2 x + 5)}{x + 3}$

Simplify the expression.

Rewrite $- (2 x + 5)$ as $- 1 (2 x + 5)$ .

$\frac{- 1 (2 x + 5)}{x + 3}$

Move the negative in front of the fraction.

$- \frac{2 x + 5}{x + 3}$

Simplify (-12x^2+6x+90)/(6x^2-54)

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