Pre-Algebra Examples

$\frac{\frac{x^{2} - 7 x + 10}{x^{2} - 8 x + 15}}{\frac{4 - x^{2}}{x^{2} + 3 x - 18}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\frac{x^{2} - 7 x + 10}{x^{2} - 8 x + 15} \cdot \frac{x^{2} + 3 x - 18}{4 - x^{2}}$

Step 2

Factor $x^{2} - 7 x + 10$ 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 $10$ and whose sum is $- 7$ .

$- 5, - 2$

Step 2.2

Write the factored form using these integers.

$\frac{(x - 5) (x - 2)}{x^{2} - 8 x + 15} \cdot \frac{x^{2} + 3 x - 18}{4 - x^{2}}$

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.

$\frac{(x - 5) (x - 2)}{(x - 5) (x - 3)} \cdot \frac{x^{2} + 3 x - 18}{4 - x^{2}}$

Step 4

Cancel the common factor of $x - 5$ .

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Step 4.1

Cancel the common factor.

$\frac{(x - 5) (x - 2)}{(x - 5) (x - 3)} \cdot \frac{x^{2} + 3 x - 18}{4 - x^{2}}$

Step 4.2

Rewrite the expression.

$\frac{x - 2}{x - 3} \cdot \frac{x^{2} + 3 x - 18}{4 - x^{2}}$

Step 5

Factor $x^{2} + 3 x - 18$ 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 $- 18$ and whose sum is $3$ .

$- 3, 6$

Step 5.2

Write the factored form using these integers.

$\frac{x - 2}{x - 3} \cdot \frac{(x - 3) (x + 6)}{4 - x^{2}}$

Step 6

Simplify the denominator.

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Step 6.1

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

$\frac{x - 2}{x - 3} \cdot \frac{(x - 3) (x + 6)}{2^{2} - x^{2}}$

Step 6.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2$ and $b = x$ .

$\frac{x - 2}{x - 3} \cdot \frac{(x - 3) (x + 6)}{(2 + x) (2 - x)}$

Step 7

Cancel the common factor of $x - 3$ .

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Step 7.1

Cancel the common factor.

$\frac{x - 2}{x - 3} \cdot \frac{(x - 3) (x + 6)}{(2 + x) (2 - x)}$

Step 7.2

Rewrite the expression.

$(x - 2) \frac{x + 6}{(2 + x) (2 - x)}$

Step 8

Multiply $x - 2$ by $\frac{x + 6}{(2 + x) (2 - x)}$ .

$\frac{(x - 2) (x + 6)}{(2 + x) (2 - x)}$

Step 9

Cancel the common factor of $x - 2$ and $2 - x$ .

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Step 9.1

Factor $- 1$ out of $x$ .

$\frac{(- 1 (- x) - 2) (x + 6)}{(2 + x) (2 - x)}$

Step 9.2

Rewrite $- 2$ as $- 1 (2)$ .

$\frac{(- 1 (- x) - 1 (2)) (x + 6)}{(2 + x) (2 - x)}$

Step 9.3

Factor $- 1$ out of $- 1 (- x) - 1 (2)$ .

$\frac{- 1 (- x + 2) (x + 6)}{(2 + x) (2 - x)}$

Step 9.4

Reorder terms.

$\frac{- 1 (- x + 2) (x + 6)}{(2 + x) (- x + 2)}$

Step 9.5

Cancel the common factor.

$\frac{- 1 (- x + 2) (x + 6)}{(2 + x) (- x + 2)}$

Step 9.6

Rewrite the expression.

$\frac{- 1 (x + 6)}{2 + x}$

Step 10

Move the negative in front of the fraction.

$- \frac{x + 6}{2 + x}$