Algebra Examples

$\frac{5 a b}{a^{2} - b^{2}} - \frac{a - b}{a + b}$

Step 1

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

$\frac{5 a b}{(a + b) (a - b)} - \frac{a - b}{a + b}$

Step 2

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

$\frac{5 a b}{(a + b) (a - b)} - \frac{a - b}{a + b} \cdot \frac{a - b}{a - b}$

Step 3

Simplify terms.

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Step 3.1

Multiply $\frac{a - b}{a + b}$ by $\frac{a - b}{a - b}$ .

$\frac{5 a b}{(a + b) (a - b)} - \frac{(a - b) (a - b)}{(a + b) (a - b)}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{5 a b - (a - b) (a - b)}{(a + b) (a - b)}$

Step 4

Simplify the numerator.

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Step 4.1

Apply the distributive property.

$\frac{5 a b + (- a - - b) (a - b)}{(a + b) (a - b)}$

Step 4.2

Multiply $- - b$ .

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Step 4.2.1

Multiply $- 1$ by $- 1$ .

$\frac{5 a b + (- a + 1 b) (a - b)}{(a + b) (a - b)}$

Step 4.2.2

Multiply $b$ by $1$ .

$\frac{5 a b + (- a + b) (a - b)}{(a + b) (a - b)}$

Step 4.3

Expand $(- a + b) (a - b)$ using the FOIL Method.

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Step 4.3.1

Apply the distributive property.

$\frac{5 a b - a (a - b) + b (a - b)}{(a + b) (a - b)}$

Step 4.3.2

Apply the distributive property.

$\frac{5 a b - a \cdot a - a (- b) + b (a - b)}{(a + b) (a - b)}$

Step 4.3.3

Apply the distributive property.

$\frac{5 a b - a \cdot a - a (- b) + b a + b (- b)}{(a + b) (a - b)}$

Step 4.4

Simplify and combine like terms.

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Step 4.4.1

Simplify each term.

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Step 4.4.1.1

Multiply $a$ by $a$ by adding the exponents.

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Step 4.4.1.1.1

Move $a$ .

$\frac{5 a b - (a \cdot a) - a (- b) + b a + b (- b)}{(a + b) (a - b)}$

Step 4.4.1.1.2

Multiply $a$ by $a$ .

$\frac{5 a b - a^{2} - a (- b) + b a + b (- b)}{(a + b) (a - b)}$

Step 4.4.1.2

Rewrite using the commutative property of multiplication.

$\frac{5 a b - a^{2} - 1 \cdot - 1 a b + b a + b (- b)}{(a + b) (a - b)}$

Step 4.4.1.3

Multiply $- 1$ by $- 1$ .

$\frac{5 a b - a^{2} + 1 a b + b a + b (- b)}{(a + b) (a - b)}$

Step 4.4.1.4

Multiply $a$ by $1$ .

$\frac{5 a b - a^{2} + a b + b a + b (- b)}{(a + b) (a - b)}$

Step 4.4.1.5

Rewrite using the commutative property of multiplication.

$\frac{5 a b - a^{2} + a b + b a - b \cdot b}{(a + b) (a - b)}$

Step 4.4.1.6

Multiply $b$ by $b$ by adding the exponents.

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Step 4.4.1.6.1

Move $b$ .

$\frac{5 a b - a^{2} + a b + b a - (b \cdot b)}{(a + b) (a - b)}$

Step 4.4.1.6.2

Multiply $b$ by $b$ .

$\frac{5 a b - a^{2} + a b + b a - b^{2}}{(a + b) (a - b)}$

Step 4.4.2

Add $a b$ and $b a$ .

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Step 4.4.2.1

Reorder $b$ and $a$ .

$\frac{5 a b - a^{2} + a b + a b - b^{2}}{(a + b) (a - b)}$

Step 4.4.2.2

Add $a b$ and $a b$ .

$\frac{5 a b - a^{2} + 2 a b - b^{2}}{(a + b) (a - b)}$

Step 4.5

Add $5 a b$ and $2 a b$ .

$\frac{- a^{2} + 7 a b - b^{2}}{(a + b) (a - b)}$

Step 5

Simplify with factoring out.

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Step 5.1

Factor $- 1$ out of $- a^{2}$ .

$\frac{- (a^{2}) + 7 a b - b^{2}}{(a + b) (a - b)}$

Step 5.2

Factor $- 1$ out of $7 a b$ .

$\frac{- (a^{2}) - (- 7 a b) - b^{2}}{(a + b) (a - b)}$

Step 5.3

Factor $- 1$ out of $- (a^{2}) - (- 7 a b)$ .

$\frac{- (a^{2} - 7 a b) - b^{2}}{(a + b) (a - b)}$

Step 5.4

Factor $- 1$ out of $- b^{2}$ .

$\frac{- (a^{2} - 7 a b) - (b^{2})}{(a + b) (a - b)}$

Step 5.5

Factor $- 1$ out of $- (a^{2} - 7 a b) - (b^{2})$ .

$\frac{- (a^{2} - 7 a b + b^{2})}{(a + b) (a - b)}$

Step 5.6

Simplify the expression.

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Step 5.6.1

Rewrite $- (a^{2} - 7 a b + b^{2})$ as $- 1 (a^{2} - 7 a b + b^{2})$ .

$\frac{- 1 (a^{2} - 7 a b + b^{2})}{(a + b) (a - b)}$

Step 5.6.2

Move the negative in front of the fraction.

$- \frac{a^{2} - 7 a b + b^{2}}{(a + b) (a - b)}$