Algebra Examples

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

Step 1

Simplify the numerator.

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Step 1.1

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

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

Step 1.2

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 $b = 2$ .

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

Step 2

Simplify with factoring out.

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Step 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.1.3

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

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

Step 2.2

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

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Step 2.2.1

Factor $2$ out of $2 x$ .

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

Step 2.2.2

Factor $2$ out of $4$ .

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

Step 2.2.3

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

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

Step 3

Factor by grouping.

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Step 3.1

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 $b = 5$ .

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Step 3.1.1

Factor $5$ out of $5 x$ .

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

Step 3.1.2

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

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

Step 3.1.3

Apply the distributive property.

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

Step 3.2

Factor out the greatest common factor from each group.

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Step 3.2.1

Group the first two terms and the last two terms.

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

Step 3.2.2

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

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

Step 3.3

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

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

Step 4

Cancel the common factor of $2$ .

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Step 4.1

Factor $2$ out of $4 x (x - 3)$ .

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

Step 4.2

Cancel the common factor.

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

Step 4.3

Rewrite the expression.

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

Step 5

Multiply $\frac{(x + 2) (x - 2)}{2 x (x - 3)}$ by $\frac{x + 2}{(- x + 2) (x - 3)}$ .

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

Add $1$ and $1$ .

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

Step 10

Raise $x - 3$ to the power of $1$ .

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

Step 11

Raise $x - 3$ to the power of $1$ .

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

Step 12

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

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

Step 13

Add $1$ and $1$ .

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

Step 14

Cancel the common factor of $x - 2$ and $- x + 2$ .

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Step 14.1

Factor $- 1$ out of $x$ .

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

Step 14.2

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

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

Step 14.3

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

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

Step 14.4

Cancel the common factor.

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

Step 14.5

Rewrite the expression.

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

Step 15

Simplify the expression.

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Step 15.1

Move $- 1$ to the left of ${(x + 2)}^{2}$ .

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

Step 15.2

Move the negative in front of the fraction.

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