Algebra Examples

$\frac{1 - \frac{3}{x + 6}}{\frac{9}{x + 6} + x}$

Step 1

Multiply the numerator and denominator of the fraction by $x + 6$ .

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Step 1.1

Multiply $\frac{1 - \frac{3}{x + 6}}{\frac{9}{x + 6} + x}$ by $\frac{x + 6}{x + 6}$ .

$\frac{x + 6}{x + 6} \cdot \frac{1 - \frac{3}{x + 6}}{\frac{9}{x + 6} + x}$

Step 1.2

Combine.

$\frac{(x + 6) (1 - \frac{3}{x + 6})}{(x + 6) (\frac{9}{x + 6} + x)}$

Step 2

Apply the distributive property.

$\frac{(x + 6) \cdot 1 + (x + 6) (- \frac{3}{x + 6})}{(x + 6) \frac{9}{x + 6} + (x + 6) x}$

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $x + 6$ .

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Step 3.1.1

Move the leading negative in $- \frac{3}{x + 6}$ into the numerator.

$\frac{(x + 6) \cdot 1 + (x + 6) \frac{- 3}{x + 6}}{(x + 6) \frac{9}{x + 6} + (x + 6) x}$

Step 3.1.2

Cancel the common factor.

$\frac{(x + 6) \cdot 1 + (x + 6) \frac{- 3}{x + 6}}{(x + 6) \frac{9}{x + 6} + (x + 6) x}$

Step 3.1.3

Rewrite the expression.

$\frac{(x + 6) \cdot 1 - 3}{(x + 6) \frac{9}{x + 6} + (x + 6) x}$

Step 3.2

Cancel the common factor of $x + 6$ .

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Step 3.2.1

Cancel the common factor.

$\frac{(x + 6) \cdot 1 - 3}{(x + 6) \frac{9}{x + 6} + (x + 6) x}$

Step 3.2.2

Rewrite the expression.

$\frac{(x + 6) \cdot 1 - 3}{9 + (x + 6) x}$

Step 4

Simplify the numerator.

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Step 4.1

Multiply $x + 6$ by $1$ .

$\frac{x + 6 - 3}{9 + (x + 6) x}$

Step 4.2

Subtract $3$ from $6$ .

$\frac{x + 3}{9 + (x + 6) x}$

Step 5

Simplify the denominator.

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Step 5.1

Apply the distributive property.

$\frac{x + 3}{9 + x \cdot x + 6 x}$

Step 5.2

Multiply $x$ by $x$ .

$\frac{x + 3}{9 + x^{2} + 6 x}$

Step 5.3

Reorder terms.

$\frac{x + 3}{x^{2} + 6 x + 9}$

Step 5.4

Factor using the perfect square rule.

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Step 5.4.1

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

$\frac{x + 3}{x^{2} + 6 x + 3^{2}}$

Step 5.4.2

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

$6 x = 2 \cdot x \cdot 3$

Step 5.4.3

Rewrite the polynomial.

$\frac{x + 3}{x^{2} + 2 \cdot x \cdot 3 + 3^{2}}$

Step 5.4.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 3$ .

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

Step 6

Cancel the common factor of $x + 3$ and ${(x + 3)}^{2}$ .

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Step 6.1

Multiply by $1$ .

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

Step 6.2

Cancel the common factors.

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Step 6.2.1

Factor $x + 3$ out of ${(x + 3)}^{2}$ .

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

Step 6.2.2

Cancel the common factor.

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

Step 6.2.3

Rewrite the expression.

$\frac{1}{x + 3}$