Algebra Examples

$\frac{x^{2} + 10 x + 25}{x - 5} \div \frac{x^{2} - 25}{5 x + 10}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{2} + 10 x + 25}{x - 5} \cdot \frac{5 x + 10}{x^{2} - 25}$

Step 2

Factor using the perfect square rule.

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Step 2.1

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

$\frac{x^{2} + 10 x + 5^{2}}{x - 5} \cdot \frac{5 x + 10}{x^{2} - 25}$

Step 2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$10 x = 2 \cdot x \cdot 5$

Step 2.3

Rewrite the polynomial.

$\frac{x^{2} + 2 \cdot x \cdot 5 + 5^{2}}{x - 5} \cdot \frac{5 x + 10}{x^{2} - 25}$

Step 2.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 5$ .

$\frac{{(x + 5)}^{2}}{x - 5} \cdot \frac{5 x + 10}{x^{2} - 25}$

Step 3

Factor $5$ out of $5 x + 10$ .

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Step 3.1

Factor $5$ out of $5 x$ .

$\frac{{(x + 5)}^{2}}{x - 5} \cdot \frac{5 (x) + 10}{x^{2} - 25}$

Step 3.2

Factor $5$ out of $10$ .

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

Step 3.3

Factor $5$ out of $5 x + 5 \cdot 2$ .

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

Step 4

Simplify the denominator.

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Step 4.1

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

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

Step 4.2

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 $b = 5$ .

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

Step 5

Cancel the common factor of $x + 5$ .

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Step 5.1

Factor $x + 5$ out of ${(x + 5)}^{2}$ .

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

Step 5.2

Cancel the common factor.

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

Step 5.3

Rewrite the expression.

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

Add $1$ and $1$ .

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

Step 11

Move $5$ to the left of $x + 5$ .

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

Step 12

Apply the distributive property.

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

Step 13

Multiply $5$ by $5$ .

$\frac{(5 x + 25) (x + 2)}{{(x - 5)}^{2}}$

Step 14

Expand $(5 x + 25) (x + 2)$ using the FOIL Method.

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Step 14.1

Apply the distributive property.

$\frac{5 x (x + 2) + 25 (x + 2)}{{(x - 5)}^{2}}$

Step 14.2

Apply the distributive property.

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

Step 14.3

Apply the distributive property.

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

Step 15

Simplify and combine like terms.

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Step 15.1

Simplify each term.

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Step 15.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 15.1.1.1

Move $x$ .

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

Step 15.1.1.2

Multiply $x$ by $x$ .

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

Step 15.1.2

Multiply $2$ by $5$ .

$\frac{5 x^{2} + 10 x + 25 x + 25 \cdot 2}{{(x - 5)}^{2}}$

Step 15.1.3

Multiply $25$ by $2$ .

$\frac{5 x^{2} + 10 x + 25 x + 50}{{(x - 5)}^{2}}$

Step 15.2

Add $10 x$ and $25 x$ .

$\frac{5 x^{2} + 35 x + 50}{{(x - 5)}^{2}}$

Step 16

Split the fraction $\frac{5 x^{2} + 35 x + 50}{{(x - 5)}^{2}}$ into two fractions.

$\frac{5 x^{2} + 35 x}{{(x - 5)}^{2}} + \frac{50}{{(x - 5)}^{2}}$

Step 17

Split the fraction $\frac{5 x^{2} + 35 x}{{(x - 5)}^{2}}$ into two fractions.

$\frac{5 x^{2}}{{(x - 5)}^{2}} + \frac{35 x}{{(x - 5)}^{2}} + \frac{50}{{(x - 5)}^{2}}$