Simplify ((x^2+13x+42)/(2x^2+28x+96)*(x^2+13x+40)/(-6x^2-30x))÷((-x-7)/-9)

$\frac{x^{2} + 13 x + 42}{2 x^{2} + 28 x + 96} \cdot \frac{x^{2} + 13 x + 40}{- 6 x^{2} - 30 x} \div \frac{- x - 7}{- 9}$

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{2} + 13 x + 42}{2 x^{2} + 28 x + 96} \cdot \frac{x^{2} + 13 x + 40}{- 6 x^{2} - 30 x} \frac{- 9}{- x - 7}$

Factor $x^{2} + 13 x + 42$ 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 $42$ and whose sum is $13$ .

$6, 7$

Write the factored form using these integers.

$\frac{(x + 6) (x + 7)}{2 x^{2} + 28 x + 96} \cdot \frac{x^{2} + 13 x + 40}{- 6 x^{2} - 30 x} \frac{- 9}{- x - 7}$

Simplify the denominator.

Factor $2$ out of $2 x^{2} + 28 x + 96$ .

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

$\frac{(x + 6) (x + 7)}{2 (x^{2}) + 28 x + 96} \cdot \frac{x^{2} + 13 x + 40}{- 6 x^{2} - 30 x} \frac{- 9}{- x - 7}$

Factor $2$ out of $28 x$ .

$\frac{(x + 6) (x + 7)}{2 (x^{2}) + 2 (14 x) + 96} \cdot \frac{x^{2} + 13 x + 40}{- 6 x^{2} - 30 x} \frac{- 9}{- x - 7}$

Factor $2$ out of $96$ .

$\frac{(x + 6) (x + 7)}{2 x^{2} + 2 (14 x) + 2 \cdot 48} \cdot \frac{x^{2} + 13 x + 40}{- 6 x^{2} - 30 x} \frac{- 9}{- x - 7}$

Factor $2$ out of $2 x^{2} + 2 (14 x)$ .

$\frac{(x + 6) (x + 7)}{2 (x^{2} + 14 x) + 2 \cdot 48} \cdot \frac{x^{2} + 13 x + 40}{- 6 x^{2} - 30 x} \frac{- 9}{- x - 7}$

Factor $2$ out of $2 (x^{2} + 14 x) + 2 \cdot 48$ .

$\frac{(x + 6) (x + 7)}{2 (x^{2} + 14 x + 48)} \cdot \frac{x^{2} + 13 x + 40}{- 6 x^{2} - 30 x} \frac{- 9}{- x - 7}$

Factor $x^{2} + 14 x + 48$ 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 $48$ and whose sum is $14$ .

$6, 8$

Write the factored form using these integers.

$\frac{(x + 6) (x + 7)}{2 ((x + 6) (x + 8))} \cdot \frac{x^{2} + 13 x + 40}{- 6 x^{2} - 30 x} \frac{- 9}{- x - 7}$

$\frac{(x + 6) (x + 7)}{2 (x + 6) (x + 8)} \cdot \frac{x^{2} + 13 x + 40}{- 6 x^{2} - 30 x} \frac{- 9}{- x - 7}$

Factor $x^{2} + 13 x + 40$ 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 $40$ and whose sum is $13$ .

$5, 8$

Write the factored form using these integers.

$\frac{(x + 6) (x + 7)}{2 (x + 6) (x + 8)} \cdot \frac{(x + 5) (x + 8)}{- 6 x^{2} - 30 x} \frac{- 9}{- x - 7}$

Simplify terms.

Factor $6 x$ out of $- 6 x^{2} - 30 x$ .

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

$\frac{(x + 6) (x + 7)}{2 (x + 6) (x + 8)} \cdot \frac{(x + 5) (x + 8)}{6 x (- x) - 30 x} \frac{- 9}{- x - 7}$

Factor $6 x$ out of $- 30 x$ .

$\frac{(x + 6) (x + 7)}{2 (x + 6) (x + 8)} \cdot \frac{(x + 5) (x + 8)}{6 x (- x) + 6 x (- 5)} \frac{- 9}{- x - 7}$

Factor $6 x$ out of $6 x (- x) + 6 x (- 5)$ .

$\frac{(x + 6) (x + 7)}{2 (x + 6) (x + 8)} \cdot \frac{(x + 5) (x + 8)}{6 x (- x - 5)} \frac{- 9}{- x - 7}$

Cancel the common factor of $x + 8$ .

Factor $x + 8$ out of $2 (x + 6) (x + 8)$ .

$\frac{(x + 6) (x + 7)}{(x + 8) (2 (x + 6))} \cdot \frac{(x + 5) (x + 8)}{6 x (- x - 5)} \frac{- 9}{- x - 7}$

Factor $x + 8$ out of $(x + 5) (x + 8)$ .

$\frac{(x + 6) (x + 7)}{(x + 8) (2 (x + 6))} \cdot \frac{(x + 8) (x + 5)}{6 x (- x - 5)} \frac{- 9}{- x - 7}$

Cancel the common factor.

$\frac{(x + 6) (x + 7)}{(x + 8) (2 (x + 6))} \cdot \frac{(x + 8) (x + 5)}{6 x (- x - 5)} \frac{- 9}{- x - 7}$

Rewrite the expression.

$\frac{(x + 6) (x + 7)}{2 (x + 6)} \cdot \frac{x + 5}{6 x (- x - 5)} \frac{- 9}{- x - 7}$

Multiply $\frac{(x + 6) (x + 7)}{2 (x + 6)}$ by $\frac{x + 5}{6 x (- x - 5)}$ .

$\frac{(x + 6) (x + 7) (x + 5)}{2 (x + 6) (6 x (- x - 5))} \cdot \frac{- 9}{- x - 7}$

Multiply $6$ by $2$ .

$\frac{(x + 6) (x + 7) (x + 5)}{12 (x + 6) (x (- x - 5))} \cdot \frac{- 9}{- x - 7}$

Cancel the common factor of $3$ .

Factor $3$ out of $12 (x + 6) (x (- x - 5))$ .

$\frac{(x + 6) (x + 7) (x + 5)}{3 (4 (x + 6) (x (- x - 5)))} \cdot \frac{- 9}{- x - 7}$

Factor $3$ out of $- 9$ .

$\frac{(x + 6) (x + 7) (x + 5)}{3 (4 (x + 6) (x (- x - 5)))} \cdot \frac{3 (- 3)}{- x - 7}$

Cancel the common factor.

$\frac{(x + 6) (x + 7) (x + 5)}{3 (4 (x + 6) (x (- x - 5)))} \cdot \frac{3 \cdot - 3}{- x - 7}$

Rewrite the expression.

$\frac{(x + 6) (x + 7) (x + 5)}{4 (x + 6) (x (- x - 5))} \cdot \frac{- 3}{- x - 7}$

Multiply $\frac{(x + 6) (x + 7) (x + 5)}{4 (x + 6) (x (- x - 5))}$ by $\frac{- 3}{- x - 7}$ .

$\frac{(x + 6) (x + 7) (x + 5) \cdot - 3}{4 (x + 6) (x (- x - 5)) (- x - 7)}$

Cancel the common factor of $x + 6$ .

Cancel the common factor.

$\frac{(x + 6) (x + 7) (x + 5) \cdot - 3}{4 (x + 6) (x (- x - 5)) (- x - 7)}$

Rewrite the expression.

$\frac{((x + 7) (x + 5)) \cdot - 3}{(4 (x (- x - 5))) (- x - 7)}$

Cancel the common factor of $x + 7$ and $- x - 7$ .

Factor $- 1$ out of $x$ .

$\frac{(- 1 (- x) + 7) (x + 5) \cdot - 3}{(4 (x (- x - 5))) (- x - 7)}$

Rewrite $7$ as $- 1 (- 7)$ .

$\frac{(- 1 (- x) - 1 (- 7)) (x + 5) \cdot - 3}{(4 (x (- x - 5))) (- x - 7)}$

Factor $- 1$ out of $- 1 (- x) - 1 (- 7)$ .

$\frac{- 1 (- x - 7) (x + 5) \cdot - 3}{(4 (x (- x - 5))) (- x - 7)}$

Cancel the common factor.

$\frac{- 1 (- x - 7) (x + 5) \cdot - 3}{4 x (- x - 5) (- x - 7)}$

Rewrite the expression.

$\frac{(- 1 (x + 5)) \cdot - 3}{4 x (- x - 5)}$

Cancel the common factor of $x + 5$ and $- x - 5$ .

Factor $- 1$ out of $x$ .

$\frac{- 1 (- 1 (- x) + 5) \cdot - 3}{4 x (- x - 5)}$

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

$\frac{- 1 (- 1 (- x) - 1 (- 5)) \cdot - 3}{4 x (- x - 5)}$

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

$\frac{- 1 (- 1 (- x - 5)) \cdot - 3}{4 x (- x - 5)}$

Cancel the common factor.

$\frac{- 1 (- 1 (- x - 5)) \cdot - 3}{4 x (- x - 5)}$

Rewrite the expression.

$\frac{- 1 \cdot (- 1) \cdot - 3}{4 x}$

Simplify the numerator.

Multiply $- 1$ by $- 1$ .

$\frac{1 \cdot - 3}{4 x}$

Multiply $- 3$ by $1$ .

$\frac{- 3}{4 x}$

Move the negative in front of the fraction.

$- \frac{3}{4 x}$

Simplify ((x^2+13x+42)/(2x^2+28x+96)*(x^2+13x+40)/(-6x^2-30x))÷((-x-7)/-9)

Related Questions

Support LCMGCF.com by disabling your adblocker.