Algebra Examples

$\frac{x^{2} - 9}{x^{2} - 25} \div \frac{x^{2} - 8 x + 15}{x^{2} + 7 x + 10}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{2} - 9}{x^{2} - 25} \cdot \frac{x^{2} + 7 x + 10}{x^{2} - 8 x + 15}$

Step 2

Simplify the numerator.

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Step 2.1

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

$\frac{x^{2} - 3^{2}}{x^{2} - 25} \cdot \frac{x^{2} + 7 x + 10}{x^{2} - 8 x + 15}$

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 $b = 3$ .

$\frac{(x + 3) (x - 3)}{x^{2} - 25} \cdot \frac{x^{2} + 7 x + 10}{x^{2} - 8 x + 15}$

Step 3

Simplify the denominator.

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Step 3.1

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

$\frac{(x + 3) (x - 3)}{x^{2} - 5^{2}} \cdot \frac{x^{2} + 7 x + 10}{x^{2} - 8 x + 15}$

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 $b = 5$ .

$\frac{(x + 3) (x - 3)}{(x + 5) (x - 5)} \cdot \frac{x^{2} + 7 x + 10}{x^{2} - 8 x + 15}$

Step 4

Factor $x^{2} + 7 x + 10$ using the AC method.

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Step 4.1

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 $7$ .

$2, 5$

Step 4.2

Write the factored form using these integers.

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

Step 5

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

$- 5, - 3$

Step 5.2

Write the factored form using these integers.

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

Step 6

Simplify terms.

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Step 6.1

Cancel the common factor of $x - 3$ .

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Step 6.1.1

Factor $x - 3$ out of $(x + 3) (x - 3)$ .

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

Step 6.1.2

Factor $x - 3$ out of $(x - 5) (x - 3)$ .

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

Step 6.1.3

Cancel the common factor.

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

Step 6.1.4

Rewrite the expression.

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

Step 6.2

Cancel the common factor of $x + 5$ .

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Step 6.2.1

Factor $x + 5$ out of $(x + 2) (x + 5)$ .

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

Step 6.2.2

Cancel the common factor.

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

Step 6.2.3

Rewrite the expression.

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

Step 6.3

Multiply $\frac{x + 3}{x - 5}$ by $\frac{x + 2}{x - 5}$ .

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

Step 7

Raise $x - 5$ to the power of $1$ .

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

Step 8

Raise $x - 5$ to the power of $1$ .

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

Step 9

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

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

Step 10

Add $1$ and $1$ .

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