Algebra Examples

$\frac{x^{2} - 8 x + 15}{x^{2} + 4 x} \div (x^{2} - x - 20)$

Step 1

Rewrite the division as a fraction.

$\frac{\frac{x^{2} - 8 x + 15}{x^{2} + 4 x}}{x^{2} - x - 20}$

Step 2

Multiply the numerator by the reciprocal of the denominator.

$\frac{x^{2} - 8 x + 15}{x^{2} + 4 x} \cdot \frac{1}{x^{2} - x - 20}$

Step 3

Factor $x^{2} - 8 x + 15$ 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 $15$ and whose sum is $- 8$ .

$- 5, - 3$

Step 3.2

Write the factored form using these integers.

$\frac{(x - 5) (x - 3)}{x^{2} + 4 x} \cdot \frac{1}{x^{2} - x - 20}$

Step 4

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

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Step 4.1

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

$\frac{(x - 5) (x - 3)}{x \cdot x + 4 x} \cdot \frac{1}{x^{2} - x - 20}$

Step 4.2

Factor $x$ out of $4 x$ .

$\frac{(x - 5) (x - 3)}{x \cdot x + x \cdot 4} \cdot \frac{1}{x^{2} - x - 20}$

Step 4.3

Factor $x$ out of $x \cdot x + x \cdot 4$ .

$\frac{(x - 5) (x - 3)}{x (x + 4)} \cdot \frac{1}{x^{2} - x - 20}$

Step 5

Factor $x^{2} - x - 20$ using the AC method.

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Step 5.1

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 $- 20$ and whose sum is $- 1$ .

$- 5, 4$

Step 5.2

Write the factored form using these integers.

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

Step 6

Simplify terms.

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Step 6.1

Cancel the common factor of $x - 5$ .

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Step 6.1.1

Cancel the common factor.

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

Step 6.1.2

Rewrite the expression.

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

Step 6.2

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

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

Step 7

Raise $x + 4$ to the power of $1$ .

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

Step 8

Raise $x + 4$ to the power of $1$ .

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

Step 9

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

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

Step 10

Add $1$ and $1$ .

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