Algebra Examples

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

Step 1

To divide by a fraction, multiply by its reciprocal.

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

Step 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 $b = q$ .

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

Step 3

Cancel the common factor of $p + q$ .

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Step 3.1

Cancel the common factor.

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

Step 3.2

Rewrite the expression.

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

Step 4

Multiply $\frac{p - q}{p + q}$ by $p - q$ .

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

Step 5

Simplify the numerator.

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

Add $1$ and $1$ .

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

Step 6

Rewrite ${(p - q)}^{2}$ as $(p - q) (p - q)$ .

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

Step 7

Expand $(p - q) (p - q)$ using the FOIL Method.

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Step 7.1

Apply the distributive property.

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

Step 7.2

Apply the distributive property.

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

Step 7.3

Apply the distributive property.

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

Step 8

Simplify and combine like terms.

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Step 8.1

Simplify each term.

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Step 8.1.1

Multiply $p$ by $p$ .

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

Step 8.1.2

Rewrite using the commutative property of multiplication.

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

Step 8.1.3

Rewrite using the commutative property of multiplication.

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

Step 8.1.4

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

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Step 8.1.4.1

Move $q$ .

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

Step 8.1.4.2

Multiply $q$ by $q$ .

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

Step 8.1.5

Multiply $- 1$ by $- 1$ .

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

Step 8.1.6

Multiply $q^{2}$ by $1$ .

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

Step 8.2

Subtract $q p$ from $- p q$ .

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Step 8.2.1

Move $q$ .

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

Step 8.2.2

Subtract $p q$ from $- p q$ .

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

Step 9

Split the fraction $\frac{p^{2} - 2 p q + q^{2}}{p + q}$ into two fractions.

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

Step 10

Split the fraction $\frac{p^{2} - 2 p q}{p + q}$ into two fractions.

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

Step 11

Move the negative in front of the fraction.

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