Pre-Algebra Examples

$\frac{\frac{x^{2} + 2 x - 35}{x^{2} - 25}}{\frac{x^{2} + 5 x - 50}{x^{2} + 4 x - 5}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\frac{x^{2} + 2 x - 35}{x^{2} - 25} \cdot \frac{x^{2} + 4 x - 5}{x^{2} + 5 x - 50}$

Step 2

Factor $x^{2} + 2 x - 35$ 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 $- 35$ and whose sum is $2$ .

$- 5, 7$

Step 2.2

Write the factored form using these integers.

$\frac{(x - 5) (x + 7)}{x^{2} - 25} \cdot \frac{x^{2} + 4 x - 5}{x^{2} + 5 x - 50}$

Step 3

Simplify the denominator.

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Step 3.1

Rewrite $25$ as $5^{2}$ .

$\frac{(x - 5) (x + 7)}{x^{2} - 5^{2}} \cdot \frac{x^{2} + 4 x - 5}{x^{2} + 5 x - 50}$

Step 3.2

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 $b = 5$ .

$\frac{(x - 5) (x + 7)}{(x + 5) (x - 5)} \cdot \frac{x^{2} + 4 x - 5}{x^{2} + 5 x - 50}$

Step 4

Cancel the common factor of $x - 5$ .

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Step 4.1

Cancel the common factor.

$\frac{(x - 5) (x + 7)}{(x + 5) (x - 5)} \cdot \frac{x^{2} + 4 x - 5}{x^{2} + 5 x - 50}$

Step 4.2

Rewrite the expression.

$\frac{x + 7}{x + 5} \cdot \frac{x^{2} + 4 x - 5}{x^{2} + 5 x - 50}$

Step 5

Factor $x^{2} + 4 x - 5$ 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 $- 5$ and whose sum is $4$ .

$- 1, 5$

Step 5.2

Write the factored form using these integers.

$\frac{x + 7}{x + 5} \cdot \frac{(x - 1) (x + 5)}{x^{2} + 5 x - 50}$

Step 6

Factor $x^{2} + 5 x - 50$ 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 $- 50$ and whose sum is $5$ .

$- 5, 10$

Step 6.2

Write the factored form using these integers.

$\frac{x + 7}{x + 5} \cdot \frac{(x - 1) (x + 5)}{(x - 5) (x + 10)}$

Step 7

Cancel the common factor of $x + 5$ .

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Step 7.1

Factor $x + 5$ out of $(x - 1) (x + 5)$ .

$\frac{x + 7}{x + 5} \cdot \frac{(x + 5) (x - 1)}{(x - 5) (x + 10)}$

Step 7.2

Cancel the common factor.

$\frac{x + 7}{x + 5} \cdot \frac{(x + 5) (x - 1)}{(x - 5) (x + 10)}$

Step 7.3

Rewrite the expression.

$(x + 7) \frac{x - 1}{(x - 5) (x + 10)}$

Step 8

Multiply $x + 7$ by $\frac{x - 1}{(x - 5) (x + 10)}$ .

$\frac{(x + 7) (x - 1)}{(x - 5) (x + 10)}$