Algebra Examples

$\frac{x^{2} - 16}{9 - x} \cdot \frac{x^{2} + x - 90}{x^{2} + 14 x + 40}$

Step 1

Simplify the numerator.

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Step 1.1

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

$\frac{x^{2} - 4^{2}}{9 - x} \cdot \frac{x^{2} + x - 90}{x^{2} + 14 x + 40}$

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 $b = 4$ .

$\frac{(x + 4) (x - 4)}{9 - x} \cdot \frac{x^{2} + x - 90}{x^{2} + 14 x + 40}$

Step 2

Factor $x^{2} + x - 90$ using the AC method.

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Step 2.1

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 $- 90$ and whose sum is $1$ .

$- 9, 10$

Step 2.2

Write the factored form using these integers.

$\frac{(x + 4) (x - 4)}{9 - x} \cdot \frac{(x - 9) (x + 10)}{x^{2} + 14 x + 40}$

Step 3

Factor $x^{2} + 14 x + 40$ 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 $40$ and whose sum is $14$ .

$4, 10$

Step 3.2

Write the factored form using these integers.

$\frac{(x + 4) (x - 4)}{9 - x} \cdot \frac{(x - 9) (x + 10)}{(x + 4) (x + 10)}$

Step 4

Simplify terms.

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Step 4.1

Cancel the common factor of $x + 4$ .

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Step 4.1.1

Cancel the common factor.

$\frac{(x + 4) (x - 4)}{9 - x} \cdot \frac{(x - 9) (x + 10)}{(x + 4) (x + 10)}$

Step 4.1.2

Rewrite the expression.

$\frac{x - 4}{9 - x} \cdot \frac{(x - 9) (x + 10)}{x + 10}$

Step 4.2

Simplify terms.

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Step 4.2.1

Multiply $\frac{x - 4}{9 - x}$ by $\frac{(x - 9) (x + 10)}{x + 10}$ .

$\frac{(x - 4) ((x - 9) (x + 10))}{(9 - x) (x + 10)}$

Step 4.2.2

Cancel the common factor of $x + 10$ .

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Step 4.2.2.1

Cancel the common factor.

$\frac{(x - 4) ((x - 9) (x + 10))}{(9 - x) (x + 10)}$

Step 4.2.2.2

Rewrite the expression.

$\frac{(x - 4) (x - 9)}{9 - x}$

Step 4.2.3

Cancel the common factor of $x - 9$ and $9 - x$ .

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Step 4.2.3.1

Factor $- 1$ out of $x$ .

$\frac{(x - 4) (- 1 (- x) - 9)}{9 - x}$

Step 4.2.3.2

Rewrite $- 9$ as $- 1 (9)$ .

$\frac{(x - 4) (- 1 (- x) - 1 (9))}{9 - x}$

Step 4.2.3.3

Factor $- 1$ out of $- 1 (- x) - 1 (9)$ .

$\frac{(x - 4) (- 1 (- x + 9))}{9 - x}$

Step 4.2.3.4

Reorder terms.

$\frac{(x - 4) (- 1 (- x + 9))}{- x + 9}$

Step 4.2.3.5

Cancel the common factor.

$\frac{(x - 4) (- 1 (- x + 9))}{- x + 9}$

Step 4.2.3.6

Divide $(x - 4) \cdot (- 1)$ by $1$ .

$(x - 4) \cdot (- 1)$

Step 4.2.4

Apply the distributive property.

$x \cdot - 1 - 4 \cdot - 1$

Step 4.2.5

Simplify the expression.

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Step 4.2.5.1

Move $- 1$ to the left of $x$ .

$- 1 \cdot x - 4 \cdot - 1$

Step 4.2.5.2

Multiply $- 4$ by $- 1$ .

$- 1 \cdot x + 4$

Step 4.3

Rewrite $- 1 x$ as $- x$ .

$- x + 4$