Algebra Examples

$\frac{\frac{x - 3}{x + 8} + \frac{x - 8}{x + 3}}{\frac{x - 3}{x + 8} - \frac{x - 8}{x + 3}}$

Step 1

Multiply the numerator and denominator of the fraction by $(x + 8) (x + 3)$ .

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Step 1.1

Multiply $\frac{\frac{x - 3}{x + 8} + \frac{x - 8}{x + 3}}{\frac{x - 3}{x + 8} - \frac{x - 8}{x + 3}}$ by $\frac{(x + 8) (x + 3)}{(x + 8) (x + 3)}$ .

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

Step 1.2

Combine.

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

Step 2

Apply the distributive property.

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

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $x + 8$ .

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Step 3.1.1

Cancel the common factor.

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

Step 3.1.2

Rewrite the expression.

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

Step 3.2

Cancel the common factor of $x + 3$ .

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Step 3.2.1

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

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

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

Step 3.3

Cancel the common factor of $x + 8$ .

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Step 3.3.1

Cancel the common factor.

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

Step 3.3.2

Rewrite the expression.

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

Step 3.4

Cancel the common factor of $x + 3$ .

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Step 3.4.1

Move the leading negative in $- \frac{x - 8}{x + 3}$ into the numerator.

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

Step 3.4.2

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

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

Step 3.4.3

Cancel the common factor.

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

Step 3.4.4

Rewrite the expression.

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

Step 4

Simplify the numerator.

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Step 4.1

Expand $(x + 3) (x - 3)$ using the FOIL Method.

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Step 4.1.1

Apply the distributive property.

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

Step 4.1.2

Apply the distributive property.

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

Step 4.1.3

Apply the distributive property.

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

Step 4.2

Combine the opposite terms in $x \cdot x + x \cdot - 3 + 3 x + 3 \cdot - 3$ .

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Step 4.2.1

Reorder the factors in the terms $x \cdot - 3$ and $3 x$ .

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

Step 4.2.2

Add $- 3 x$ and $3 x$ .

$\frac{x \cdot x + 0 + 3 \cdot - 3 + (x + 8) (x - 8)}{(x + 3) (x - 3) + (x + 8) (- (x - 8))}$

Step 4.2.3

Add $x \cdot x$ and $0$ .

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

Step 4.3

Simplify each term.

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Step 4.3.1

Multiply $x$ by $x$ .

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

Step 4.3.2

Multiply $3$ by $- 3$ .

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

Step 4.4

Expand $(x + 8) (x - 8)$ using the FOIL Method.

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Step 4.4.1

Apply the distributive property.

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

Step 4.4.2

Apply the distributive property.

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

Step 4.4.3

Apply the distributive property.

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

Step 4.5

Combine the opposite terms in $x \cdot x + x \cdot - 8 + 8 x + 8 \cdot - 8$ .

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Step 4.5.1

Reorder the factors in the terms $x \cdot - 8$ and $8 x$ .

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

Step 4.5.2

Add $- 8 x$ and $8 x$ .

$\frac{x^{2} - 9 + x \cdot x + 0 + 8 \cdot - 8}{(x + 3) (x - 3) + (x + 8) (- (x - 8))}$

Step 4.5.3

Add $x \cdot x$ and $0$ .

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

Step 4.6

Simplify each term.

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Step 4.6.1

Multiply $x$ by $x$ .

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

Step 4.6.2

Multiply $8$ by $- 8$ .

$\frac{x^{2} - 9 + x^{2} - 64}{(x + 3) (x - 3) + (x + 8) (- (x - 8))}$

Step 4.7

Add $x^{2}$ and $x^{2}$ .

$\frac{2 x^{2} - 9 - 64}{(x + 3) (x - 3) + (x + 8) (- (x - 8))}$

Step 4.8

Subtract $64$ from $- 9$ .

$\frac{2 x^{2} - 73}{(x + 3) (x - 3) + (x + 8) (- (x - 8))}$

Step 5

Simplify the denominator.

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Step 5.1

Expand $(x + 3) (x - 3)$ using the FOIL Method.

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Step 5.1.1

Apply the distributive property.

$\frac{2 x^{2} - 73}{x (x - 3) + 3 (x - 3) + (x + 8) (- (x - 8))}$

Step 5.1.2

Apply the distributive property.

$\frac{2 x^{2} - 73}{x \cdot x + x \cdot - 3 + 3 (x - 3) + (x + 8) (- (x - 8))}$

Step 5.1.3

Apply the distributive property.

$\frac{2 x^{2} - 73}{x \cdot x + x \cdot - 3 + 3 x + 3 \cdot - 3 + (x + 8) (- (x - 8))}$

Step 5.2

Combine the opposite terms in $x \cdot x + x \cdot - 3 + 3 x + 3 \cdot - 3$ .

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Step 5.2.1

Reorder the factors in the terms $x \cdot - 3$ and $3 x$ .

$\frac{2 x^{2} - 73}{x \cdot x - 3 x + 3 x + 3 \cdot - 3 + (x + 8) (- (x - 8))}$

Step 5.2.2

Add $- 3 x$ and $3 x$ .

$\frac{2 x^{2} - 73}{x \cdot x + 0 + 3 \cdot - 3 + (x + 8) (- (x - 8))}$

Step 5.2.3

Add $x \cdot x$ and $0$ .

$\frac{2 x^{2} - 73}{x \cdot x + 3 \cdot - 3 + (x + 8) (- (x - 8))}$

Step 5.3

Simplify each term.

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Step 5.3.1

Multiply $x$ by $x$ .

$\frac{2 x^{2} - 73}{x^{2} + 3 \cdot - 3 + (x + 8) (- (x - 8))}$

Step 5.3.2

Multiply $3$ by $- 3$ .

$\frac{2 x^{2} - 73}{x^{2} - 9 + (x + 8) (- (x - 8))}$

Step 5.4

Apply the distributive property.

$\frac{2 x^{2} - 73}{x^{2} - 9 + (x + 8) (- x - - 8)}$

Step 5.5

Multiply $- 1$ by $- 8$ .

$\frac{2 x^{2} - 73}{x^{2} - 9 + (x + 8) (- x + 8)}$

Step 5.6

Expand $(x + 8) (- x + 8)$ using the FOIL Method.

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Step 5.6.1

Apply the distributive property.

$\frac{2 x^{2} - 73}{x^{2} - 9 + x (- x + 8) + 8 (- x + 8)}$

Step 5.6.2

Apply the distributive property.

$\frac{2 x^{2} - 73}{x^{2} - 9 + x (- x) + x \cdot 8 + 8 (- x + 8)}$

Step 5.6.3

Apply the distributive property.

$\frac{2 x^{2} - 73}{x^{2} - 9 + x (- x) + x \cdot 8 + 8 (- x) + 8 \cdot 8}$

Step 5.7

Simplify and combine like terms.

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Step 5.7.1

Simplify each term.

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Step 5.7.1.1

Rewrite using the commutative property of multiplication.

$\frac{2 x^{2} - 73}{x^{2} - 9 - x \cdot x + x \cdot 8 + 8 (- x) + 8 \cdot 8}$

Step 5.7.1.2

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

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Step 5.7.1.2.1

Move $x$ .

$\frac{2 x^{2} - 73}{x^{2} - 9 - (x \cdot x) + x \cdot 8 + 8 (- x) + 8 \cdot 8}$

Step 5.7.1.2.2

Multiply $x$ by $x$ .

$\frac{2 x^{2} - 73}{x^{2} - 9 - x^{2} + x \cdot 8 + 8 (- x) + 8 \cdot 8}$

Step 5.7.1.3

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

$\frac{2 x^{2} - 73}{x^{2} - 9 - x^{2} + 8 \cdot x + 8 (- x) + 8 \cdot 8}$

Step 5.7.1.4

Multiply $- 1$ by $8$ .

$\frac{2 x^{2} - 73}{x^{2} - 9 - x^{2} + 8 x - 8 x + 8 \cdot 8}$

Step 5.7.1.5

Multiply $8$ by $8$ .

$\frac{2 x^{2} - 73}{x^{2} - 9 - x^{2} + 8 x - 8 x + 64}$

Step 5.7.2

Subtract $8 x$ from $8 x$ .

$\frac{2 x^{2} - 73}{x^{2} - 9 - x^{2} + 0 + 64}$

Step 5.7.3

Add $- x^{2}$ and $0$ .

$\frac{2 x^{2} - 73}{x^{2} - 9 - x^{2} + 64}$

Step 5.8

Subtract $x^{2}$ from $x^{2}$ .

$\frac{2 x^{2} - 73}{0 - 9 + 64}$

Step 5.9

Subtract $9$ from $0$ .

$\frac{2 x^{2} - 73}{- 9 + 64}$

Step 5.10

Add $- 9$ and $64$ .

$\frac{2 x^{2} - 73}{55}$