Pre-Algebra Examples

\frac{7 x + 1}{x^{2} - 36} + \frac{3 x}{x + 6}

$\frac{7 x + 1}{x^{2} - 36} + \frac{3 x}{x + 6}$

Step 1

Simplify the denominator.

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Step 1.1

Rewrite

36

$36$ as

6^{2}

$6^{2}$ .

\frac{7 x + 1}{x^{2} - 6^{2}} + \frac{3 x}{x + 6}

$\frac{7 x + 1}{x^{2} - 6^{2}} + \frac{3 x}{x + 6}$

Step 1.2

Since both terms are perfect squares, factor using the difference of squares formula,

a^{2} - b^{2} = (a + b) (a - b)

$a^{2} - b^{2} = (a + b) (a - b)$ where

a = x

$a = x$ and

b = 6

$b = 6$ .

\frac{7 x + 1}{(x + 6) (x - 6)} + \frac{3 x}{x + 6}

$\frac{7 x + 1}{(x + 6) (x - 6)} + \frac{3 x}{x + 6}$

\frac{7 x + 1}{(x + 6) (x - 6)} + \frac{3 x}{x + 6}

$\frac{7 x + 1}{(x + 6) (x - 6)} + \frac{3 x}{x + 6}$

Step 2

To write

\frac{3 x}{x + 6}

$\frac{3 x}{x + 6}$ as a fraction with a common denominator, multiply by

\frac{x - 6}{x - 6}

$\frac{x - 6}{x - 6}$ .

\frac{7 x + 1}{(x + 6) (x - 6)} + \frac{3 x}{x + 6} \cdot \frac{x - 6}{x - 6}

$\frac{7 x + 1}{(x + 6) (x - 6)} + \frac{3 x}{x + 6} \cdot \frac{x - 6}{x - 6}$

Step 3

Simplify terms.

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Step 3.1

Multiply

\frac{3 x}{x + 6}

$\frac{3 x}{x + 6}$ by

\frac{x - 6}{x - 6}

$\frac{x - 6}{x - 6}$ .

\frac{7 x + 1}{(x + 6) (x - 6)} + \frac{3 x (x - 6)}{(x + 6) (x - 6)}

$\frac{7 x + 1}{(x + 6) (x - 6)} + \frac{3 x (x - 6)}{(x + 6) (x - 6)}$

Step 3.2

Combine the numerators over the common denominator.

\frac{7 x + 1 + 3 x (x - 6)}{(x + 6) (x - 6)}

$\frac{7 x + 1 + 3 x (x - 6)}{(x + 6) (x - 6)}$

\frac{7 x + 1 + 3 x (x - 6)}{(x + 6) (x - 6)}

$\frac{7 x + 1 + 3 x (x - 6)}{(x + 6) (x - 6)}$

Step 4

Simplify the numerator.

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Step 4.1

Apply the distributive property.

\frac{7 x + 1 + 3 x \cdot x + 3 x \cdot - 6}{(x + 6) (x - 6)}

$\frac{7 x + 1 + 3 x \cdot x + 3 x \cdot - 6}{(x + 6) (x - 6)}$

Step 4.2

Multiply

x

$x$ by

x

$x$ by adding the exponents.

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Step 4.2.1

Move

x

$x$ .

\frac{7 x + 1 + 3 (x \cdot x) + 3 x \cdot - 6}{(x + 6) (x - 6)}

$\frac{7 x + 1 + 3 (x \cdot x) + 3 x \cdot - 6}{(x + 6) (x - 6)}$

Step 4.2.2

Multiply

x

$x$ by

x

$x$ .

\frac{7 x + 1 + 3 x^{2} + 3 x \cdot - 6}{(x + 6) (x - 6)}

$\frac{7 x + 1 + 3 x^{2} + 3 x \cdot - 6}{(x + 6) (x - 6)}$

\frac{7 x + 1 + 3 x^{2} + 3 x \cdot - 6}{(x + 6) (x - 6)}

$\frac{7 x + 1 + 3 x^{2} + 3 x \cdot - 6}{(x + 6) (x - 6)}$

Step 4.3

Multiply

- 6

$- 6$ by

3

$3$ .

\frac{7 x + 1 + 3 x^{2} - 18 x}{(x + 6) (x - 6)}

$\frac{7 x + 1 + 3 x^{2} - 18 x}{(x + 6) (x - 6)}$

Step 4.4

Subtract

18 x

$18 x$ from

7 x

$7 x$ .

\frac{- 11 x + 1 + 3 x^{2}}{(x + 6) (x - 6)}

$\frac{- 11 x + 1 + 3 x^{2}}{(x + 6) (x - 6)}$

Step 4.5

Reorder terms.

\frac{3 x^{2} - 11 x + 1}{(x + 6) (x - 6)}

$\frac{3 x^{2} - 11 x + 1}{(x + 6) (x - 6)}$

\frac{3 x^{2} - 11 x + 1}{(x + 6) (x - 6)}

$\frac{3 x^{2} - 11 x + 1}{(x + 6) (x - 6)}$