Algebra Examples

$(\frac{p q}{p^{2} - q^{2}} + \frac{q}{q - p}) \div (p - q + \frac{4 q^{2} - p^{2}}{p + q})$

Step 1

Rewrite the division as a fraction.

$\frac{\frac{p q}{p^{2} - q^{2}} + \frac{q}{q - p}}{p - q + \frac{4 q^{2} - p^{2}}{p + q}}$

Step 2

Multiply the numerator and denominator of the fraction by $(p^{2} - q^{2}) (q - p) (p + q)$ .

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Step 2.1

Multiply $\frac{\frac{p q}{p^{2} - q^{2}} + \frac{q}{q - p}}{p - q + \frac{4 q^{2} - p^{2}}{p + q}}$ by $\frac{(p^{2} - q^{2}) (q - p) (p + q)}{(p^{2} - q^{2}) (q - p) (p + q)}$ .

$\frac{(p^{2} - q^{2}) (q - p) (p + q)}{(p^{2} - q^{2}) (q - p) (p + q)} \cdot \frac{\frac{p q}{p^{2} - q^{2}} + \frac{q}{q - p}}{p - q + \frac{4 q^{2} - p^{2}}{p + q}}$

Step 2.2

Combine.

$\frac{(p^{2} - q^{2}) (q - p) (p + q) (\frac{p q}{p^{2} - q^{2}} + \frac{q}{q - p})}{(p^{2} - q^{2}) (q - p) (p + q) (p - q + \frac{4 q^{2} - p^{2}}{p + q})}$

Step 3

Apply the distributive property.

$\frac{(p^{2} - q^{2}) (q - p) (p + q) \frac{p q}{p^{2} - q^{2}} + (p^{2} - q^{2}) (q - p) (p + q) \frac{q}{q - p}}{(p^{2} - q^{2}) (q - p) (p + q) p + (p^{2} - q^{2}) (q - p) (p + q) (- q) + (p^{2} - q^{2}) (q - p) (p + q) \frac{4 q^{2} - p^{2}}{p + q}}$

Step 4

Simplify by cancelling.

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Step 4.1

Cancel the common factor of $p^{2} - q^{2}$ .

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Step 4.1.1

Factor $p^{2} - q^{2}$ out of $(p^{2} - q^{2}) (q - p) (p + q)$ .

$\frac{(p^{2} - q^{2}) ((q - p) (p + q)) \frac{p q}{p^{2} - q^{2}} + (p^{2} - q^{2}) (q - p) (p + q) \frac{q}{q - p}}{(p^{2} - q^{2}) (q - p) (p + q) p + (p^{2} - q^{2}) (q - p) (p + q) (- q) + (p^{2} - q^{2}) (q - p) (p + q) \frac{4 q^{2} - p^{2}}{p + q}}$

Step 4.1.2

Cancel the common factor.

Step 4.1.3

Rewrite the expression.

$\frac{(q - p) (p + q) (p q) + (p^{2} - q^{2}) (q - p) (p + q) \frac{q}{q - p}}{(p^{2} - q^{2}) (q - p) (p + q) p + (p^{2} - q^{2}) (q - p) (p + q) (- q) + (p^{2} - q^{2}) (q - p) (p + q) \frac{4 q^{2} - p^{2}}{p + q}}$

Step 4.2

Cancel the common factor of $q - p$ .

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Step 4.2.1

Factor $q - p$ out of $(p^{2} - q^{2}) (q - p) (p + q)$ .

$\frac{(q - p) (p + q) (p q) + (q - p) ((p^{2} - q^{2}) (p + q)) \frac{q}{q - p}}{(p^{2} - q^{2}) (q - p) (p + q) p + (p^{2} - q^{2}) (q - p) (p + q) (- q) + (p^{2} - q^{2}) (q - p) (p + q) \frac{4 q^{2} - p^{2}}{p + q}}$

Step 4.2.2

Cancel the common factor.

Step 4.2.3

Rewrite the expression.

$\frac{(q - p) (p + q) (p q) + (p^{2} - q^{2}) (p + q) q}{(p^{2} - q^{2}) (q - p) (p + q) p + (p^{2} - q^{2}) (q - p) (p + q) (- q) + (p^{2} - q^{2}) (q - p) (p + q) \frac{4 q^{2} - p^{2}}{p + q}}$

Step 4.3

Cancel the common factor of $p + q$ .

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Step 4.3.1

Factor $p + q$ out of $(p^{2} - q^{2}) (q - p) (p + q)$ .

$\frac{(q - p) (p + q) (p q) + (p^{2} - q^{2}) (p + q) q}{(p^{2} - q^{2}) (q - p) (p + q) p + (p^{2} - q^{2}) (q - p) (p + q) (- q) + (p + q) ((p^{2} - q^{2}) (q - p)) \frac{4 q^{2} - p^{2}}{p + q}}$

Step 4.3.2

Cancel the common factor.

$\frac{(q - p) (p + q) (p q) + (p^{2} - q^{2}) (p + q) q}{(p^{2} - q^{2}) (q - p) (p + q) p + (p^{2} - q^{2}) (q - p) (p + q) (- q) + (p + q) ((p^{2} - q^{2}) (q - p)) \frac{4 q^{2} - p^{2}}{p + q}}$

Step 4.3.3

Rewrite the expression.

$\frac{(q - p) (p + q) (p q) + (p^{2} - q^{2}) (p + q) q}{(p^{2} - q^{2}) (q - p) (p + q) p + (p^{2} - q^{2}) (q - p) (p + q) (- q) + (p^{2} - q^{2}) (q - p) (4 q^{2} - p^{2})}$

Step 5

Simplify the numerator.

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Step 5.1

Factor $(p + q) q$ out of $(q - p) (p + q) (p q) + (p^{2} - q^{2}) (p + q) q$ .

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Step 5.1.1

Factor $(p + q) q$ out of $(q - p) (p + q) (p q)$ .

$\frac{(p + q) q ((q - p) (p)) + (p^{2} - q^{2}) (p + q) q}{(p^{2} - q^{2}) (q - p) (p + q) p + (p^{2} - q^{2}) (q - p) (p + q) (- q) + (p^{2} - q^{2}) (q - p) (4 q^{2} - p^{2})}$

Step 5.1.2

Factor $(p + q) q$ out of $(p^{2} - q^{2}) (p + q) q$ .

$\frac{(p + q) q ((q - p) (p)) + (p + q) q (p^{2} - q^{2})}{(p^{2} - q^{2}) (q - p) (p + q) p + (p^{2} - q^{2}) (q - p) (p + q) (- q) + (p^{2} - q^{2}) (q - p) (4 q^{2} - p^{2})}$

Step 5.1.3

Factor $(p + q) q$ out of $(p + q) q ((q - p) (p)) + (p + q) q (p^{2} - q^{2})$ .

$\frac{(p + q) q ((q - p) (p) + p^{2} - q^{2})}{(p^{2} - q^{2}) (q - p) (p + q) p + (p^{2} - q^{2}) (q - p) (p + q) (- q) + (p^{2} - q^{2}) (q - p) (4 q^{2} - p^{2})}$

Step 5.2

Apply the distributive property.

$\frac{(p + q) q (q p - p \cdot p + p^{2} - q^{2})}{(p^{2} - q^{2}) (q - p) (p + q) p + (p^{2} - q^{2}) (q - p) (p + q) (- q) + (p^{2} - q^{2}) (q - p) (4 q^{2} - p^{2})}$

Step 5.3

Multiply $p$ by $p$ by adding the exponents.

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Step 5.3.1

Move $p$ .

$\frac{(p + q) q (q p - (p \cdot p) + p^{2} - q^{2})}{(p^{2} - q^{2}) (q - p) (p + q) p + (p^{2} - q^{2}) (q - p) (p + q) (- q) + (p^{2} - q^{2}) (q - p) (4 q^{2} - p^{2})}$

Step 5.3.2

Multiply $p$ by $p$ .

$\frac{(p + q) q (q p - p^{2} + p^{2} - q^{2})}{(p^{2} - q^{2}) (q - p) (p + q) p + (p^{2} - q^{2}) (q - p) (p + q) (- q) + (p^{2} - q^{2}) (q - p) (4 q^{2} - p^{2})}$

Step 5.4

Add $- p^{2}$ and $p^{2}$ .

$\frac{(p + q) q (q p + 0 - q^{2})}{(p^{2} - q^{2}) (q - p) (p + q) p + (p^{2} - q^{2}) (q - p) (p + q) (- q) + (p^{2} - q^{2}) (q - p) (4 q^{2} - p^{2})}$

Step 5.5

Add $q p$ and $0$ .

$\frac{(p + q) q (q p - q^{2})}{(p^{2} - q^{2}) (q - p) (p + q) p + (p^{2} - q^{2}) (q - p) (p + q) (- q) + (p^{2} - q^{2}) (q - p) (4 q^{2} - p^{2})}$

Step 5.6

Factor $q$ out of $q p - q^{2}$ .

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Step 5.6.1

Factor $q$ out of $q p$ .

$\frac{(p + q) q (q (p) - q^{2})}{(p^{2} - q^{2}) (q - p) (p + q) p + (p^{2} - q^{2}) (q - p) (p + q) (- q) + (p^{2} - q^{2}) (q - p) (4 q^{2} - p^{2})}$

Step 5.6.2

Factor $q$ out of $- q^{2}$ .

$\frac{(p + q) q (q (p) + q (- q))}{(p^{2} - q^{2}) (q - p) (p + q) p + (p^{2} - q^{2}) (q - p) (p + q) (- q) + (p^{2} - q^{2}) (q - p) (4 q^{2} - p^{2})}$

Step 5.6.3

Factor $q$ out of $q (p) + q (- q)$ .

$\frac{(p + q) q (q (p - q))}{(p^{2} - q^{2}) (q - p) (p + q) p + (p^{2} - q^{2}) (q - p) (p + q) (- q) + (p^{2} - q^{2}) (q - p) (4 q^{2} - p^{2})}$

$\frac{(p + q) q \cdot q (p - q)}{(p^{2} - q^{2}) (q - p) (p + q) p + (p^{2} - q^{2}) (q - p) (p + q) (- q) + (p^{2} - q^{2}) (q - p) (4 q^{2} - p^{2})}$

Step 5.7

Combine exponents.

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Step 5.7.1

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

$\frac{(p + q) (q^{1} q) (p - q)}{(p^{2} - q^{2}) (q - p) (p + q) p + (p^{2} - q^{2}) (q - p) (p + q) (- q) + (p^{2} - q^{2}) (q - p) (4 q^{2} - p^{2})}$

Step 5.7.2

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

$\frac{(p + q) (q^{1} q^{1}) (p - q)}{(p^{2} - q^{2}) (q - p) (p + q) p + (p^{2} - q^{2}) (q - p) (p + q) (- q) + (p^{2} - q^{2}) (q - p) (4 q^{2} - p^{2})}$

Step 5.7.3

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

$\frac{(p + q) q^{1 + 1} (p - q)}{(p^{2} - q^{2}) (q - p) (p + q) p + (p^{2} - q^{2}) (q - p) (p + q) (- q) + (p^{2} - q^{2}) (q - p) (4 q^{2} - p^{2})}$

Step 5.7.4

Add $1$ and $1$ .

$\frac{(p + q) q^{2} (p - q)}{(p^{2} - q^{2}) (q - p) (p + q) p + (p^{2} - q^{2}) (q - p) (p + q) (- q) + (p^{2} - q^{2}) (q - p) (4 q^{2} - p^{2})}$

Step 6

Simplify the denominator.

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Step 6.1

Factor $(p^{2} - q^{2}) (q - p)$ out of $(p^{2} - q^{2}) (q - p) (p + q) p + (p^{2} - q^{2}) (q - p) (p + q) (- q) + (p^{2} - q^{2}) (q - p) (4 q^{2} - p^{2})$ .

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Step 6.1.1

Factor $(p^{2} - q^{2}) (q - p)$ out of $(p^{2} - q^{2}) (q - p) (p + q) p$ .

$\frac{(p + q) q^{2} (p - q)}{(p^{2} - q^{2}) (q - p) ((p + q) p) + (p^{2} - q^{2}) (q - p) (p + q) (- q) + (p^{2} - q^{2}) (q - p) (4 q^{2} - p^{2})}$

Step 6.1.2

Factor $(p^{2} - q^{2}) (q - p)$ out of $(p^{2} - q^{2}) (q - p) (p + q) (- q)$ .

$\frac{(p + q) q^{2} (p - q)}{(p^{2} - q^{2}) (q - p) ((p + q) p) + (p^{2} - q^{2}) (q - p) ((p + q) (- q)) + (p^{2} - q^{2}) (q - p) (4 q^{2} - p^{2})}$

Step 6.1.3

Factor $(p^{2} - q^{2}) (q - p)$ out of $(p^{2} - q^{2}) (q - p) ((p + q) p) + (p^{2} - q^{2}) (q - p) ((p + q) (- q))$ .

$\frac{(p + q) q^{2} (p - q)}{(p^{2} - q^{2}) (q - p) ((p + q) p + (p + q) (- q)) + (p^{2} - q^{2}) (q - p) (4 q^{2} - p^{2})}$

Step 6.1.4

Factor $(p^{2} - q^{2}) (q - p)$ out of $(p^{2} - q^{2}) (q - p) ((p + q) p + (p + q) (- q)) + (p^{2} - q^{2}) (q - p) (4 q^{2} - p^{2})$ .

$\frac{(p + q) q^{2} (p - q)}{(p^{2} - q^{2}) (q - p) ((p + q) p + (p + q) (- q) + 4 q^{2} - p^{2})}$

Step 6.2

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

$\frac{(p + q) q^{2} (p - q)}{(p + q) (p - q) (q - p) ((p + q) p + (p + q) (- q) + 4 q^{2} - p^{2})}$

Step 6.3

Combine exponents.

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Step 6.3.1

Factor $- 1$ out of $p$ .

$\frac{(p + q) q^{2} (p - q)}{(p + q) (- 1 (- p) - q) (q - p) ((p + q) p + (p + q) (- q) + 4 q^{2} - p^{2})}$

Step 6.3.2

Factor $- 1$ out of $- q$ .

$\frac{(p + q) q^{2} (p - q)}{(p + q) (- 1 (- p) - (q)) (q - p) ((p + q) p + (p + q) (- q) + 4 q^{2} - p^{2})}$

Step 6.3.3

Factor $- 1$ out of $- 1 (- p) - (q)$ .

$\frac{(p + q) q^{2} (p - q)}{(p + q) (- 1 (- p + q)) (q - p) ((p + q) p + (p + q) (- q) + 4 q^{2} - p^{2})}$

Step 6.3.4

Reorder terms.

$\frac{(p + q) q^{2} (p - q)}{(p + q) \cdot - 1 (- p + q) (- p + q) ((p + q) p + (p + q) (- q) + 4 q^{2} - p^{2})}$

Step 6.3.5

Raise $- p + q$ to the power of $1$ .

$\frac{(p + q) q^{2} (p - q)}{(p + q) \cdot - 1 ({(- p + q)}^{1} (- p + q)) ((p + q) p + (p + q) (- q) + 4 q^{2} - p^{2})}$

Step 6.3.6

Raise $- p + q$ to the power of $1$ .

$\frac{(p + q) q^{2} (p - q)}{(p + q) \cdot - 1 ({(- p + q)}^{1} {(- p + q)}^{1}) ((p + q) p + (p + q) (- q) + 4 q^{2} - p^{2})}$

Step 6.3.7

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

$\frac{(p + q) q^{2} (p - q)}{(p + q) \cdot - 1 {(- p + q)}^{1 + 1} ((p + q) p + (p + q) (- q) + 4 q^{2} - p^{2})}$

Step 6.3.8

Add $1$ and $1$ .

$\frac{(p + q) q^{2} (p - q)}{(p + q) \cdot - 1 {(- p + q)}^{2} ((p + q) p + (p + q) (- q) + 4 q^{2} - p^{2})}$

Step 6.4

Factor out negative.

$\frac{(p + q) q^{2} (p - q)}{- (p + q) {(- p + q)}^{2} ((p + q) p + (p + q) (- q) + 4 q^{2} - p^{2})}$

Step 6.5

Simplify each term.

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Step 6.5.1

Apply the distributive property.

$\frac{(p + q) q^{2} (p - q)}{- (p + q) {(- p + q)}^{2} (p \cdot p + q p + (p + q) (- q) + 4 q^{2} - p^{2})}$

Step 6.5.2

Multiply $p$ by $p$ .

$\frac{(p + q) q^{2} (p - q)}{- (p + q) {(- p + q)}^{2} (p^{2} + q p + (p + q) (- q) + 4 q^{2} - p^{2})}$

Step 6.5.3

Apply the distributive property.

$\frac{(p + q) q^{2} (p - q)}{- (p + q) {(- p + q)}^{2} (p^{2} + q p + p (- q) + q (- q) + 4 q^{2} - p^{2})}$

Step 6.5.4

Rewrite using the commutative property of multiplication.

$\frac{(p + q) q^{2} (p - q)}{- (p + q) {(- p + q)}^{2} (p^{2} + q p - p q + q (- q) + 4 q^{2} - p^{2})}$

Step 6.5.5

Rewrite using the commutative property of multiplication.

$\frac{(p + q) q^{2} (p - q)}{- (p + q) {(- p + q)}^{2} (p^{2} + q p - p q - q \cdot q + 4 q^{2} - p^{2})}$

Step 6.5.6

Multiply $q$ by $q$ by adding the exponents.

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Step 6.5.6.1

Move $q$ .

$\frac{(p + q) q^{2} (p - q)}{- (p + q) {(- p + q)}^{2} (p^{2} + q p - p q - (q \cdot q) + 4 q^{2} - p^{2})}$

Step 6.5.6.2

Multiply $q$ by $q$ .

$\frac{(p + q) q^{2} (p - q)}{- (p + q) {(- p + q)}^{2} (p^{2} + q p - p q - q^{2} + 4 q^{2} - p^{2})}$

Step 6.6

Combine the opposite terms in $p^{2} + q p - p q - q^{2} + 4 q^{2} - p^{2}$ .

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Step 6.6.1

Reorder the factors in the terms $q p$ and $- p q$ .

$\frac{(p + q) q^{2} (p - q)}{- (p + q) {(- p + q)}^{2} (p^{2} + p q - p q - q^{2} + 4 q^{2} - p^{2})}$

Step 6.6.2

Subtract $p q$ from $p q$ .

$\frac{(p + q) q^{2} (p - q)}{- (p + q) {(- p + q)}^{2} (p^{2} + 0 - q^{2} + 4 q^{2} - p^{2})}$

Step 6.6.3

Add $p^{2}$ and $0$ .

$\frac{(p + q) q^{2} (p - q)}{- (p + q) {(- p + q)}^{2} (p^{2} - q^{2} + 4 q^{2} - p^{2})}$

Step 6.6.4

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

$\frac{(p + q) q^{2} (p - q)}{- (p + q) {(- p + q)}^{2} (- q^{2} + 4 q^{2} + 0)}$

Step 6.6.5

Add $- q^{2} + 4 q^{2}$ and $0$ .

$\frac{(p + q) q^{2} (p - q)}{- (p + q) {(- p + q)}^{2} (- q^{2} + 4 q^{2})}$

Step 6.7

Add $- q^{2}$ and $4 q^{2}$ .

$\frac{(p + q) q^{2} (p - q)}{- (p + q) {(- p + q)}^{2} \cdot 3 q^{2}}$

Step 6.8

Multiply $3$ by $- 1$ .

$\frac{(p + q) q^{2} (p - q)}{- 3 (p + q) {(- p + q)}^{2} q^{2}}$

Step 7

Simplify terms.

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Step 7.1

Cancel the common factor of $p + q$ .

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Step 7.1.1

Cancel the common factor.

$\frac{(p + q) q^{2} (p - q)}{- 3 (p + q) {(- p + q)}^{2} q^{2}}$

Step 7.1.2

Rewrite the expression.

$\frac{q^{2} (p - q)}{- 3 {(- p + q)}^{2} q^{2}}$

Step 7.2

Cancel the common factor of $q^{2}$ .

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Step 7.2.1

Cancel the common factor.

$\frac{q^{2} (p - q)}{- 3 {(- p + q)}^{2} q^{2}}$

Step 7.2.2

Rewrite the expression.

$\frac{p - q}{- 3 {(- p + q)}^{2}}$

Step 7.3

Cancel the common factor of $p - q$ and ${(- p + q)}^{2}$ .

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Step 7.3.1

Factor $- 1$ out of $p$ .

$\frac{- 1 (- p) - q}{- 3 {(- p + q)}^{2}}$

Step 7.3.2

Factor $- 1$ out of $- q$ .

$\frac{- 1 (- p) - (q)}{- 3 {(- p + q)}^{2}}$

Step 7.3.3

Factor $- 1$ out of $- 1 (- p) - (q)$ .

$\frac{- 1 (- p + q)}{- 3 {(- p + q)}^{2}}$

Step 7.3.4

Factor $- p + q$ out of $- 1 (- p + q)$ .

$\frac{(- p + q) \cdot - 1}{- 3 {(- p + q)}^{2}}$

Step 7.3.5

Cancel the common factors.

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Step 7.3.5.1

Factor $- p + q$ out of $- 3 {(- p + q)}^{2}$ .

$\frac{(- p + q) \cdot - 1}{(- p + q) (- 3 (- p + q))}$

Step 7.3.5.2

Cancel the common factor.

$\frac{(- p + q) \cdot - 1}{(- p + q) (- 3 (- p + q))}$

Step 7.3.5.3

Rewrite the expression.

$\frac{- 1}{- 3 (- p + q)}$

Step 7.4

Dividing two negative values results in a positive value.

$\frac{1}{(3) (- p + q)}$

Step 7.5

Factor $- 1$ out of $- p$ .

$\frac{1}{3 (- (p) + q)}$

Step 7.6

Factor $- 1$ out of $q$ .

$\frac{1}{3 (- (p) - 1 (- q))}$

Step 7.7

Factor $- 1$ out of $- (p) - 1 (- q)$ .

$\frac{1}{3 (- (p - q))}$

Step 7.8

Rewrite negatives.

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Step 7.8.1

Rewrite $- (p - q)$ as $- 1 (p - q)$ .

$\frac{1}{3 (- 1 (p - q))}$

Step 7.8.2

Move the negative in front of the fraction.

$- \frac{1}{3 (p - q)}$