Simplify ((2a^2-25a+72)/(4a^2-22a+18))÷((8a+8)/(2a-2))

$\frac{2 a^{2} - 25 a + 72}{4 a^{2} - 22 a + 18} \div \frac{8 a + 8}{2 a - 2}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{2 a^{2} - 25 a + 72}{4 a^{2} - 22 a + 18} \cdot \frac{2 a - 2}{8 a + 8}$

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 $a \cdot c = 2 \cdot 72 = 144$ and whose sum is $b = - 25$ .

Factor $- 25$ out of $- 25 a$ .

$\frac{2 a^{2} - 25 (a) + 72}{4 a^{2} - 22 a + 18} \cdot \frac{2 a - 2}{8 a + 8}$

Rewrite $- 25$ as $- 9$ plus $- 16$

$\frac{2 a^{2} + (- 9 - 16) a + 72}{4 a^{2} - 22 a + 18} \cdot \frac{2 a - 2}{8 a + 8}$

Apply the distributive property.

$\frac{2 a^{2} - 9 a - 16 a + 72}{4 a^{2} - 22 a + 18} \cdot \frac{2 a - 2}{8 a + 8}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(2 a^{2} - 9 a) - 16 a + 72}{4 a^{2} - 22 a + 18} \cdot \frac{2 a - 2}{8 a + 8}$

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

$\frac{a (2 a - 9) - 8 (2 a - 9)}{4 a^{2} - 22 a + 18} \cdot \frac{2 a - 2}{8 a + 8}$

Factor the polynomial by factoring out the greatest common factor, $2 a - 9$ .

$\frac{(2 a - 9) (a - 8)}{4 a^{2} - 22 a + 18} \cdot \frac{2 a - 2}{8 a + 8}$

Step 3

Simplify the denominator.

Factor $2$ out of $4 a^{2} - 22 a + 18$ .

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

$\frac{(2 a - 9) (a - 8)}{2 (2 a^{2}) - 22 a + 18} \cdot \frac{2 a - 2}{8 a + 8}$

Factor $2$ out of $- 22 a$ .

$\frac{(2 a - 9) (a - 8)}{2 (2 a^{2}) + 2 (- 11 a) + 18} \cdot \frac{2 a - 2}{8 a + 8}$

Factor $2$ out of $18$ .

$\frac{(2 a - 9) (a - 8)}{2 (2 a^{2}) + 2 (- 11 a) + 2 (9)} \cdot \frac{2 a - 2}{8 a + 8}$

Factor $2$ out of $2 (2 a^{2}) + 2 (- 11 a)$ .

$\frac{(2 a - 9) (a - 8)}{2 (2 a^{2} - 11 a) + 2 (9)} \cdot \frac{2 a - 2}{8 a + 8}$

Factor $2$ out of $2 (2 a^{2} - 11 a) + 2 (9)$ .

$\frac{(2 a - 9) (a - 8)}{2 (2 a^{2} - 11 a + 9)} \cdot \frac{2 a - 2}{8 a + 8}$

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 $a \cdot c = 2 \cdot 9 = 18$ and whose sum is $b = - 11$ .

Factor $- 11$ out of $- 11 a$ .

$\frac{(2 a - 9) (a - 8)}{2 (2 a^{2} - 11 (a) + 9)} \cdot \frac{2 a - 2}{8 a + 8}$

Rewrite $- 11$ as $- 2$ plus $- 9$

$\frac{(2 a - 9) (a - 8)}{2 (2 a^{2} + (- 2 - 9) a + 9)} \cdot \frac{2 a - 2}{8 a + 8}$

Apply the distributive property.

$\frac{(2 a - 9) (a - 8)}{2 (2 a^{2} - 2 a - 9 a + 9)} \cdot \frac{2 a - 2}{8 a + 8}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(2 a - 9) (a - 8)}{2 ((2 a^{2} - 2 a) - 9 a + 9)} \cdot \frac{2 a - 2}{8 a + 8}$

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

$\frac{(2 a - 9) (a - 8)}{2 (2 a (a - 1) - 9 (a - 1))} \cdot \frac{2 a - 2}{8 a + 8}$

Factor the polynomial by factoring out the greatest common factor, $a - 1$ .

$\frac{(2 a - 9) (a - 8)}{2 ((a - 1) (2 a - 9))} \cdot \frac{2 a - 2}{8 a + 8}$

$\frac{(2 a - 9) (a - 8)}{2 (a - 1) (2 a - 9)} \cdot \frac{2 a - 2}{8 a + 8}$

Step 4

Simplify terms.

Cancel the common factor of $2 a - 9$ .

Cancel the common factor.

$\frac{(2 a - 9) (a - 8)}{2 (a - 1) (2 a - 9)} \cdot \frac{2 a - 2}{8 a + 8}$

Rewrite the expression.

$\frac{a - 8}{2 (a - 1)} \cdot \frac{2 a - 2}{8 a + 8}$

Cancel the common factor of $2 a - 2$ and $8 a + 8$ .

Factor $2$ out of $2 a$ .

$\frac{a - 8}{2 (a - 1)} \cdot \frac{2 (a) - 2}{8 a + 8}$

Factor $2$ out of $- 2$ .

$\frac{a - 8}{2 (a - 1)} \cdot \frac{2 (a) + 2 (- 1)}{8 a + 8}$

Factor $2$ out of $2 (a) + 2 (- 1)$ .

$\frac{a - 8}{2 (a - 1)} \cdot \frac{2 (a - 1)}{8 a + 8}$

Cancel the common factors.

Factor $2$ out of $8 a$ .

$\frac{a - 8}{2 (a - 1)} \cdot \frac{2 (a - 1)}{2 (4 a) + 8}$

Factor $2$ out of $8$ .

$\frac{a - 8}{2 (a - 1)} \cdot \frac{2 (a - 1)}{2 (4 a) + 2 (4)}$

Factor $2$ out of $2 (4 a) + 2 (4)$ .

$\frac{a - 8}{2 (a - 1)} \cdot \frac{2 (a - 1)}{2 (4 a + 4)}$

Cancel the common factor.

$\frac{a - 8}{2 (a - 1)} \cdot \frac{2 (a - 1)}{2 (4 a + 4)}$

Rewrite the expression.

$\frac{a - 8}{2 (a - 1)} \cdot \frac{a - 1}{4 a + 4}$

Cancel the common factor of $a - 1$ .

Factor $a - 1$ out of $2 (a - 1)$ .

$\frac{a - 8}{(a - 1) \cdot 2} \cdot \frac{a - 1}{4 a + 4}$

Cancel the common factor.

$\frac{a - 8}{(a - 1) \cdot 2} \cdot \frac{a - 1}{4 a + 4}$

Rewrite the expression.

$\frac{a - 8}{2} \cdot \frac{1}{4 a + 4}$

Multiply $\frac{a - 8}{2}$ by $\frac{1}{4 a + 4}$ .

$\frac{a - 8}{2 (4 a + 4)}$

Step 5

Simplify the denominator.

Factor $4$ out of $4 a + 4$ .

Factor $4$ out of $4 a$ .

$\frac{a - 8}{2 (4 (a) + 4)}$

Factor $4$ out of $4$ .

$\frac{a - 8}{2 (4 (a) + 4 (1))}$

Factor $4$ out of $4 (a) + 4 (1)$ .

$\frac{a - 8}{2 (4 (a + 1))}$

$\frac{a - 8}{2 \cdot 4 (a + 1)}$

Multiply $2$ by $4$ .

$\frac{a - 8}{8 (a + 1)}$

Simplify ((2a^2-25a+72)/(4a^2-22a+18))÷((8a+8)/(2a-2))

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