Simplify (4y^2+16y-468)/(567-7y^2)

$\frac{4 y^{2} + 16 y - 468}{567 - 7 y^{2}}$

Step 1

Simplify the numerator.

Factor $4$ out of $4 y^{2} + 16 y - 468$ .

Factor $4$ out of $4 y^{2}$ .

$\frac{4 (y^{2}) + 16 y - 468}{567 - 7 y^{2}}$

Factor $4$ out of $16 y$ .

$\frac{4 (y^{2}) + 4 (4 y) - 468}{567 - 7 y^{2}}$

Factor $4$ out of $- 468$ .

$\frac{4 y^{2} + 4 (4 y) + 4 \cdot - 117}{567 - 7 y^{2}}$

Factor $4$ out of $4 y^{2} + 4 (4 y)$ .

$\frac{4 (y^{2} + 4 y) + 4 \cdot - 117}{567 - 7 y^{2}}$

Factor $4$ out of $4 (y^{2} + 4 y) + 4 \cdot - 117$ .

$\frac{4 (y^{2} + 4 y - 117)}{567 - 7 y^{2}}$

Factor $y^{2} + 4 y - 117$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 117$ and whose sum is $4$ .

$- 9, 13$

Write the factored form using these integers.

$\frac{4 ((y - 9) (y + 13))}{567 - 7 y^{2}}$

$\frac{4 (y - 9) (y + 13)}{567 - 7 y^{2}}$

Step 2

Simplify the denominator.

Factor $7$ out of $567 - 7 y^{2}$ .

Factor $7$ out of $567$ .

$\frac{4 (y - 9) (y + 13)}{7 (81) - 7 y^{2}}$

Factor $7$ out of $- 7 y^{2}$ .

$\frac{4 (y - 9) (y + 13)}{7 (81) + 7 (- y^{2})}$

Factor $7$ out of $7 (81) + 7 (- y^{2})$ .

$\frac{4 (y - 9) (y + 13)}{7 (81 - y^{2})}$

Rewrite $81$ as $9^{2}$ .

$\frac{4 (y - 9) (y + 13)}{7 (9^{2} - y^{2})}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 9$ and $b = y$ .

$\frac{4 (y - 9) (y + 13)}{7 (9 + y) (9 - y)}$

Step 3

Reduce the expression by cancelling the common factors.

Cancel the common factor of $y - 9$ and $9 - y$ .

Factor $- 1$ out of $y$ .

$\frac{4 (- 1 (- y) - 9) (y + 13)}{7 (9 + y) (9 - y)}$

Rewrite $- 9$ as $- 1 (9)$ .

$\frac{4 (- 1 (- y) - 1 (9)) (y + 13)}{7 (9 + y) (9 - y)}$

Factor $- 1$ out of $- 1 (- y) - 1 (9)$ .

$\frac{4 (- 1 (- y + 9)) (y + 13)}{7 (9 + y) (9 - y)}$

Reorder terms.

$\frac{4 (- 1 (- y + 9)) (y + 13)}{7 (9 + y) (- y + 9)}$

Cancel the common factor.

$\frac{4 (- 1 (- y + 9)) (y + 13)}{7 (9 + y) (- y + 9)}$

Rewrite the expression.

$\frac{4 \cdot (- 1) (y + 13)}{7 (9 + y)}$

Simplify the expression.

Multiply $4$ by $- 1$ .

$\frac{- 4 (y + 13)}{7 (9 + y)}$

Move the negative in front of the fraction.

$- \frac{4 (y + 13)}{7 (9 + y)}$

Simplify (4y^2+16y-468)/(567-7y^2)

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