Algebra Examples

$\frac{1 - \frac{x^{2}}{9}}{\frac{x + 8}{15} + \frac{1}{x}}$

Step 1

Multiply the numerator and denominator of the fraction by $45 x$ .

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Step 1.1

Multiply $\frac{1 - \frac{x^{2}}{9}}{\frac{x + 8}{15} + \frac{1}{x}}$ by $\frac{45 x}{45 x}$ .

$\frac{45 x}{45 x} \cdot \frac{1 - \frac{x^{2}}{9}}{\frac{x + 8}{15} + \frac{1}{x}}$

Step 1.2

Combine.

$\frac{45 x (1 - \frac{x^{2}}{9})}{45 x (\frac{x + 8}{15} + \frac{1}{x})}$

Step 2

Apply the distributive property.

$\frac{45 x \cdot 1 + 45 x (- \frac{x^{2}}{9})}{45 x \frac{x + 8}{15} + 45 x \frac{1}{x}}$

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $9$ .

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Step 3.1.1

Move the leading negative in $- \frac{x^{2}}{9}$ into the numerator.

$\frac{45 x \cdot 1 + 45 x \frac{- x^{2}}{9}}{45 x \frac{x + 8}{15} + 45 x \frac{1}{x}}$

Step 3.1.2

Factor $9$ out of $45 x$ .

$\frac{45 x \cdot 1 + 9 (5 x) \frac{- x^{2}}{9}}{45 x \frac{x + 8}{15} + 45 x \frac{1}{x}}$

Step 3.1.3

Cancel the common factor.

$\frac{45 x \cdot 1 + 9 (5 x) \frac{- x^{2}}{9}}{45 x \frac{x + 8}{15} + 45 x \frac{1}{x}}$

Step 3.1.4

Rewrite the expression.

$\frac{45 x \cdot 1 + 5 x (- x^{2})}{45 x \frac{x + 8}{15} + 45 x \frac{1}{x}}$

Step 3.2

Multiply $- 1$ by $5$ .

$\frac{45 x \cdot 1 - 5 x \cdot x^{2}}{45 x \frac{x + 8}{15} + 45 x \frac{1}{x}}$

Step 3.3

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 3.3.1

Move $x^{2}$ .

$\frac{45 x \cdot 1 - 5 (x^{2} x)}{45 x \frac{x + 8}{15} + 45 x \frac{1}{x}}$

Step 3.3.2

Multiply $x^{2}$ by $x$ .

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Step 3.3.2.1

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

$\frac{45 x \cdot 1 - 5 (x^{2} x^{1})}{45 x \frac{x + 8}{15} + 45 x \frac{1}{x}}$

Step 3.3.2.2

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

$\frac{45 x \cdot 1 - 5 x^{2 + 1}}{45 x \frac{x + 8}{15} + 45 x \frac{1}{x}}$

Step 3.3.3

Add $2$ and $1$ .

$\frac{45 x \cdot 1 - 5 x^{3}}{45 x \frac{x + 8}{15} + 45 x \frac{1}{x}}$

Step 3.4

Cancel the common factor of $15$ .

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Step 3.4.1

Factor $15$ out of $45 x$ .

$\frac{45 x \cdot 1 - 5 x^{3}}{15 (3 x) \frac{x + 8}{15} + 45 x \frac{1}{x}}$

Step 3.4.2

Cancel the common factor.

$\frac{45 x \cdot 1 - 5 x^{3}}{15 (3 x) \frac{x + 8}{15} + 45 x \frac{1}{x}}$

Step 3.4.3

Rewrite the expression.

$\frac{45 x \cdot 1 - 5 x^{3}}{3 x (x + 8) + 45 x \frac{1}{x}}$

Step 3.5

Cancel the common factor of $x$ .

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Step 3.5.1

Factor $x$ out of $45 x$ .

$\frac{45 x \cdot 1 - 5 x^{3}}{3 x (x + 8) + x \cdot 45 \frac{1}{x}}$

Step 3.5.2

Cancel the common factor.

$\frac{45 x \cdot 1 - 5 x^{3}}{3 x (x + 8) + x \cdot 45 \frac{1}{x}}$

Step 3.5.3

Rewrite the expression.

$\frac{45 x \cdot 1 - 5 x^{3}}{3 x (x + 8) + 45}$

Step 4

Simplify the numerator.

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Step 4.1

Factor $5 x$ out of $45 x \cdot 1 - 5 x^{3}$ .

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Step 4.1.1

Factor $5 x$ out of $45 x \cdot 1$ .

$\frac{5 x (9 \cdot 1) - 5 x^{3}}{3 x (x + 8) + 45}$

Step 4.1.2

Factor $5 x$ out of $- 5 x^{3}$ .

$\frac{5 x (9 \cdot 1) + 5 x (- x^{2})}{3 x (x + 8) + 45}$

Step 4.1.3

Factor $5 x$ out of $5 x (9 \cdot 1) + 5 x (- x^{2})$ .

$\frac{5 x (9 \cdot 1 - x^{2})}{3 x (x + 8) + 45}$

Step 4.2

Multiply $9$ by $1$ .

$\frac{5 x (9 - x^{2})}{3 x (x + 8) + 45}$

Step 4.3

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

$\frac{5 x (3^{2} - x^{2})}{3 x (x + 8) + 45}$

Step 4.4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 3$ and $b = x$ .

$\frac{5 x (3 + x) (3 - x)}{3 x (x + 8) + 45}$

Step 5

Simplify the denominator.

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Step 5.1

Factor $3$ out of $3 x (x + 8) + 45$ .

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Step 5.1.1

Factor $3$ out of $3 x (x + 8)$ .

$\frac{5 x (3 + x) (3 - x)}{3 (x (x + 8)) + 45}$

Step 5.1.2

Factor $3$ out of $45$ .

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

Step 5.1.3

Factor $3$ out of $3 (x (x + 8)) + 3 \cdot 15$ .

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

Step 5.2

Apply the distributive property.

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

Step 5.3

Multiply $x$ by $x$ .

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

Step 5.4

Move $8$ to the left of $x$ .

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

Step 5.5

Factor $x^{2} + 8 x + 15$ using the AC method.

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Step 5.5.1

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 $8$ .

$3, 5$

Step 5.5.2

Write the factored form using these integers.

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

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

Step 6

Cancel the common factor of $3 + x$ and $x + 3$ .

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Step 6.1

Reorder terms.

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

Step 6.2

Cancel the common factor.

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

Step 6.3

Rewrite the expression.

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