Simplify ((x^2-4)/(4x^2-12x))÷((2x+4)/(-x^2+5x-6))

$\frac{x^{2} - 4}{4 x^{2} - 12 x} \div \frac{2 x + 4}{- x^{2} + 5 x - 6}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{2} - 4}{4 x^{2} - 12 x} \cdot \frac{- x^{2} + 5 x - 6}{2 x + 4}$

Step 2

Simplify the numerator.

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

$\frac{x^{2} - 2^{2}}{4 x^{2} - 12 x} \cdot \frac{- x^{2} + 5 x - 6}{2 x + 4}$

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 $b = 2$ .

$\frac{(x + 2) (x - 2)}{4 x^{2} - 12 x} \cdot \frac{- x^{2} + 5 x - 6}{2 x + 4}$

Step 3

Factor $4 x$ out of $4 x^{2} - 12 x$ .

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

$\frac{(x + 2) (x - 2)}{4 x (x) - 12 x} \cdot \frac{- x^{2} + 5 x - 6}{2 x + 4}$

Factor $4 x$ out of $- 12 x$ .

$\frac{(x + 2) (x - 2)}{4 x (x) + 4 x (- 3)} \cdot \frac{- x^{2} + 5 x - 6}{2 x + 4}$

Factor $4 x$ out of $4 x (x) + 4 x (- 3)$ .

$\frac{(x + 2) (x - 2)}{4 x (x - 3)} \cdot \frac{- x^{2} + 5 x - 6}{2 x + 4}$

Step 4

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 $a \cdot c = - 1 \cdot - 6 = 6$ and whose sum is $b = 5$ .

Factor $5$ out of $5 x$ .

$\frac{(x + 2) (x - 2)}{4 x (x - 3)} \cdot \frac{- x^{2} + 5 (x) - 6}{2 x + 4}$

Rewrite $5$ as $2$ plus $3$

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

Apply the distributive property.

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

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

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

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

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

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

Step 5

Factor $2$ out of $2 x + 4$ .

Factor $2$ out of $2 x$ .

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

Factor $2$ out of $4$ .

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

Factor $2$ out of $2 x + 2 \cdot 2$ .

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

Step 6

Cancel the common factor of $x + 2$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 7

Cancel the common factor of $x - 3$ .

Factor $x - 3$ out of $4 x (x - 3)$ .

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

Factor $x - 3$ out of $(- x + 2) (x - 3)$ .

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

Cancel the common factor.

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

Rewrite the expression.

$\frac{x - 2}{4 x} \cdot \frac{- x + 2}{2}$

Step 8

Multiply $\frac{x - 2}{4 x}$ by $\frac{- x + 2}{2}$ .

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

Step 9

Factor $- 1$ out of $x$ .

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

Step 10

Rewrite $- 2$ as $- 1 (2)$ .

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

Step 11

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

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

Step 12

Raise $- x + 2$ to the power of $1$ .

$\frac{- 1 ({(- x + 2)}^{1} (- x + 2))}{4 x \cdot 2}$

Step 13

Raise $- x + 2$ to the power of $1$ .

$\frac{- 1 ({(- x + 2)}^{1} {(- x + 2)}^{1})}{4 x \cdot 2}$

Step 14

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{- 1 {(- x + 2)}^{1 + 1}}{4 x \cdot 2}$

Step 15

Add $1$ and $1$ .

$\frac{- 1 {(- x + 2)}^{2}}{4 x \cdot 2}$

Step 16

Multiply $2$ by $4$ .

$\frac{- 1 {(- x + 2)}^{2}}{8 x}$

Step 17

Move the negative in front of the fraction.

$- \frac{{(- x + 2)}^{2}}{8 x}$

Simplify ((x^2-4)/(4x^2-12x))÷((2x+4)/(-x^2+5x-6))

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