Pre-Algebra Examples

$\frac{8 x v}{x^{2} - 4 v^{2}} + \frac{x - 2 v}{x + 2 v}$

Step 1

Simplify the denominator.

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Step 1.1

Rewrite $4 v^{2}$ as ${(2 v)}^{2}$ .

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

Step 1.2

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

$\frac{8 x v}{(x + 2 v) (x - (2 v))} + \frac{x - 2 v}{x + 2 v}$

Step 1.3

Multiply $2$ by $- 1$ .

$\frac{8 x v}{(x + 2 v) (x - 2 v)} + \frac{x - 2 v}{x + 2 v}$

Step 2

To write $\frac{x - 2 v}{x + 2 v}$ as a fraction with a common denominator, multiply by $\frac{x - 2 v}{x - 2 v}$ .

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

Step 3

Simplify terms.

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Step 3.1

Multiply $\frac{x - 2 v}{x + 2 v}$ by $\frac{x - 2 v}{x - 2 v}$ .

$\frac{8 x v}{(x + 2 v) (x - 2 v)} + \frac{(x - 2 v) (x - 2 v)}{(x + 2 v) (x - 2 v)}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{8 x v + (x - 2 v) (x - 2 v)}{(x + 2 v) (x - 2 v)}$

Step 4

Simplify the numerator.

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Step 4.1

Expand $(x - 2 v) (x - 2 v)$ using the FOIL Method.

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Step 4.1.1

Apply the distributive property.

$\frac{8 x v + x (x - 2 v) - 2 v (x - 2 v)}{(x + 2 v) (x - 2 v)}$

Step 4.1.2

Apply the distributive property.

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

Step 4.1.3

Apply the distributive property.

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

Step 4.2

Simplify and combine like terms.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Multiply $x$ by $x$ .

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

Step 4.2.1.2

Rewrite using the commutative property of multiplication.

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

Step 4.2.1.3

Rewrite using the commutative property of multiplication.

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

Step 4.2.1.4

Multiply $v$ by $v$ by adding the exponents.

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Step 4.2.1.4.1

Move $v$ .

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

Step 4.2.1.4.2

Multiply $v$ by $v$ .

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

Step 4.2.1.5

Multiply $- 2$ by $- 2$ .

$\frac{8 x v + x^{2} - 2 x v - 2 v x + 4 v^{2}}{(x + 2 v) (x - 2 v)}$

Step 4.2.2

Subtract $2 v x$ from $- 2 x v$ .

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Step 4.2.2.1

Move $x$ .

$\frac{8 x v + x^{2} - 2 v x - 2 v x + 4 v^{2}}{(x + 2 v) (x - 2 v)}$

Step 4.2.2.2

Subtract $2 v x$ from $- 2 v x$ .

$\frac{8 x v + x^{2} - 4 v x + 4 v^{2}}{(x + 2 v) (x - 2 v)}$

Step 4.3

Subtract $4 v x$ from $8 x v$ .

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Step 4.3.1

Move $x$ .

$\frac{x^{2} + 8 v x - 4 v x + 4 v^{2}}{(x + 2 v) (x - 2 v)}$

Step 4.3.2

Subtract $4 v x$ from $8 v x$ .

$\frac{x^{2} + 4 v x + 4 v^{2}}{(x + 2 v) (x - 2 v)}$

Step 4.4

Factor using the perfect square rule.

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Step 4.4.1

Rewrite $4 v^{2}$ as ${(2 v)}^{2}$ .

$\frac{x^{2} + 4 v x + {(2 v)}^{2}}{(x + 2 v) (x - 2 v)}$

Step 4.4.2

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

$4 v x = 2 \cdot x \cdot (2 v)$

Step 4.4.3

Rewrite the polynomial.

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

Step 4.4.4

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

$\frac{{(x + 2 v)}^{2}}{(x + 2 v) (x - 2 v)}$

Step 5

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

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Step 5.1

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

$\frac{(x + 2 v) (x + 2 v)}{(x + 2 v) (x - 2 v)}$

Step 5.2

Cancel the common factors.

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Step 5.2.1

Cancel the common factor.

$\frac{(x + 2 v) (x + 2 v)}{(x + 2 v) (x - 2 v)}$

Step 5.2.2

Rewrite the expression.

$\frac{x + 2 v}{x - 2 v}$