Pre-Algebra Examples

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

Step 1

Simplify the denominator.

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Step 1.1

Rewrite $36$ as $6^{2}$ .

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

Step 1.2

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 $b = 6$ .

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

Step 2

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

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

Step 3

Combine the numerators over the common denominator.

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

Step 4

Simplify each term.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Factor $x$ out of $x (x + 6) - x^{2}$ .

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Step 4.1.1.1

Factor $x$ out of $- x^{2}$ .

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

Step 4.1.1.2

Factor $x$ out of $x (x + 6) + x (- x)$ .

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

Step 4.1.2

Subtract $x$ from $x$ .

$\frac{x (0 + 6)}{x + 6} - \frac{x}{(x + 6) (x - 6)}$

Step 4.1.3

Add $0$ and $6$ .

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

Step 4.2

Move $6$ to the left of $x$ .

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

Step 5

To write $\frac{6 x}{x + 6}$ as a fraction with a common denominator, multiply by $\frac{x - 6}{x - 6}$ .

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

Step 6

Simplify terms.

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Step 6.1

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

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

Step 6.2

Combine the numerators over the common denominator.

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

Step 7

Simplify the numerator.

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Step 7.1

Factor $x$ out of $6 x (x - 6) - x$ .

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Step 7.1.1

Factor $x$ out of $6 x (x - 6)$ .

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

Step 7.1.2

Factor $x$ out of $- x$ .

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

Step 7.1.3

Factor $x$ out of $x (6 (x - 6)) + x \cdot - 1$ .

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

Step 7.2

Apply the distributive property.

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

Step 7.3

Multiply $6$ by $- 6$ .

$\frac{x (6 x - 36 - 1)}{(x + 6) (x - 6)}$

Step 7.4

Subtract $1$ from $- 36$ .

$\frac{x (6 x - 37)}{(x + 6) (x - 6)}$