Algebra Examples

$\frac{\frac{x^{2} - 8 x + 16}{x^{2} + 4 x - 21}}{\frac{- 2 x + 8}{x + 7}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\frac{x^{2} - 8 x + 16}{x^{2} + 4 x - 21} \cdot \frac{x + 7}{- 2 x + 8}$

Step 2

Factor using the perfect square rule.

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Step 2.1

Rewrite $16$ as $4^{2}$ .

$\frac{x^{2} - 8 x + 4^{2}}{x^{2} + 4 x - 21} \cdot \frac{x + 7}{- 2 x + 8}$

Step 2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$8 x = 2 \cdot x \cdot 4$

Step 2.3

Rewrite the polynomial.

$\frac{x^{2} - 2 \cdot x \cdot 4 + 4^{2}}{x^{2} + 4 x - 21} \cdot \frac{x + 7}{- 2 x + 8}$

Step 2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 4$ .

$\frac{{(x - 4)}^{2}}{x^{2} + 4 x - 21} \cdot \frac{x + 7}{- 2 x + 8}$

Step 3

Factor $x^{2} + 4 x - 21$ using the AC method.

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Step 3.1

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 $- 21$ and whose sum is $4$ .

$- 3, 7$

Step 3.2

Write the factored form using these integers.

$\frac{{(x - 4)}^{2}}{(x - 3) (x + 7)} \cdot \frac{x + 7}{- 2 x + 8}$

Step 4

Simplify terms.

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Step 4.1

Cancel the common factor of $x + 7$ .

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Step 4.1.1

Factor $x + 7$ out of $(x - 3) (x + 7)$ .

$\frac{{(x - 4)}^{2}}{(x + 7) (x - 3)} \cdot \frac{x + 7}{- 2 x + 8}$

Step 4.1.2

Cancel the common factor.

$\frac{{(x - 4)}^{2}}{(x + 7) (x - 3)} \cdot \frac{x + 7}{- 2 x + 8}$

Step 4.1.3

Rewrite the expression.

$\frac{{(x - 4)}^{2}}{x - 3} \cdot \frac{1}{- 2 x + 8}$

Step 4.2

Multiply $\frac{{(x - 4)}^{2}}{x - 3}$ by $\frac{1}{- 2 x + 8}$ .

$\frac{{(x - 4)}^{2}}{(x - 3) (- 2 x + 8)}$

Step 5

Factor $2$ out of $- 2 x + 8$ .

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Step 5.1

Factor $2$ out of $- 2 x$ .

$\frac{{(x - 4)}^{2}}{(x - 3) (2 (- x) + 8)}$

Step 5.2

Factor $2$ out of $8$ .

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

Step 5.3

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

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

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

Step 6

Cancel the common factor of ${(x - 4)}^{2}$ and $- x + 4$ .

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Step 6.1

Factor $- 1$ out of $x$ .

$\frac{{(- 1 (- x) - 4)}^{2}}{(x - 3) \cdot 2 (- x + 4)}$

Step 6.2

Rewrite $- 4$ as $- 1 (4)$ .

$\frac{{(- 1 (- x) - 1 (4))}^{2}}{(x - 3) \cdot 2 (- x + 4)}$

Step 6.3

Factor $- 1$ out of $- 1 (- x) - 1 (4)$ .

$\frac{{(- 1 (- x + 4))}^{2}}{(x - 3) \cdot 2 (- x + 4)}$

Step 6.4

Apply the product rule to $- 1 (- x + 4)$ .

$\frac{{(- 1)}^{2} {(- x + 4)}^{2}}{(x - 3) \cdot 2 (- x + 4)}$

Step 6.5

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

$\frac{1 {(- x + 4)}^{2}}{(x - 3) \cdot 2 (- x + 4)}$

Step 6.6

Multiply ${(- x + 4)}^{2}$ by $1$ .

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

Step 6.7

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

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

Step 6.8

Cancel the common factors.

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Step 6.8.1

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

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

Step 6.8.2

Cancel the common factor.

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

Step 6.8.3

Rewrite the expression.

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

Step 7

Move $2$ to the left of $x - 3$ .

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

Step 8

Factor $- 1$ out of $- x$ .

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

Step 9

Rewrite $4$ as $- 1 (- 4)$ .

$\frac{- (x) - 1 (- 4)}{2 (x - 3)}$

Step 10

Factor $- 1$ out of $- (x) - 1 (- 4)$ .

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

Step 11

Simplify the expression.

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Step 11.1

Rewrite $- (x - 4)$ as $- 1 (x - 4)$ .

$\frac{- 1 (x - 4)}{2 (x - 3)}$

Step 11.2

Move the negative in front of the fraction.

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