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

Write each expression with a common denominator of $(a + b) (a - b)$ , by multiplying each by an appropriate factor of $1$ .

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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

Reorder the factors of $(a - b) (a + b)$ .

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

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

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Step 5.1.1

Apply the distributive property.

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

Step 5.1.2

Apply the distributive property.

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

Step 5.1.3

Apply the distributive property.

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

Step 5.2

Simplify and combine like terms.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Multiply $a$ by $a$ .

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

Step 5.2.1.2

Multiply $b$ by $b$ .

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

Step 5.2.2

Add $a b$ and $b a$ .

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Step 5.2.2.1

Reorder $b$ and $a$ .

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

Step 5.2.2.2

Add $a b$ and $a b$ .

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

Step 5.3

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

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