Pre-Algebra Examples

$\frac{\frac{x^{2} - 9}{x^{2} + 5 x - 24}}{\frac{x^{2} - 5 x - 24}{x - 3}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\frac{x^{2} - 9}{x^{2} + 5 x - 24} \cdot \frac{x - 3}{x^{2} - 5 x - 24}$

Step 2

Simplify the numerator.

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Step 2.1

Rewrite $9$ as $3^{2}$ .

$\frac{x^{2} - 3^{2}}{x^{2} + 5 x - 24} \cdot \frac{x - 3}{x^{2} - 5 x - 24}$

Step 2.2

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 $b = 3$ .

$\frac{(x + 3) (x - 3)}{x^{2} + 5 x - 24} \cdot \frac{x - 3}{x^{2} - 5 x - 24}$

Step 3

Factor $x^{2} + 5 x - 24$ 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 $- 24$ and whose sum is $5$ .

$- 3, 8$

Step 3.2

Write the factored form using these integers.

$\frac{(x + 3) (x - 3)}{(x - 3) (x + 8)} \cdot \frac{x - 3}{x^{2} - 5 x - 24}$

Step 4

Cancel the common factor of $x - 3$ .

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Step 4.1

Cancel the common factor.

$\frac{(x + 3) (x - 3)}{(x - 3) (x + 8)} \cdot \frac{x - 3}{x^{2} - 5 x - 24}$

Step 4.2

Rewrite the expression.

$\frac{(x + 3) (x - 3)}{x + 8} \cdot \frac{1}{x^{2} - 5 x - 24}$

Step 5

Multiply $\frac{(x + 3) (x - 3)}{x + 8}$ by $\frac{1}{x^{2} - 5 x - 24}$ .

$\frac{(x + 3) (x - 3)}{(x + 8) (x^{2} - 5 x - 24)}$

Step 6

Factor $x^{2} - 5 x - 24$ using the AC method.

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Step 6.1

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 $- 24$ and whose sum is $- 5$ .

$- 8, 3$

Step 6.2

Write the factored form using these integers.

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

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

Step 7

Cancel the common factor of $x + 3$ .

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Step 7.1

Cancel the common factor.

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

Step 7.2

Rewrite the expression.

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

Step 8

Split the fraction $\frac{x - 3}{(x + 8) (x - 8)}$ into two fractions.

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

Step 9

Move the negative in front of the fraction.

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