Algebra Examples

$(6 y - \frac{1}{2} q^{2}) (6 y + \frac{1}{2} q^{2})$

Step 1

Combine fractions.

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Step 1.1

Combine $q^{2}$ and $\frac{1}{2}$ .

$(6 y - \frac{q^{2}}{2}) (6 y + \frac{1}{2} q^{2})$

Step 1.2

Combine $\frac{1}{2}$ and $q^{2}$ .

$(6 y - \frac{q^{2}}{2}) (6 y + \frac{q^{2}}{2})$

Step 2

Expand $(6 y - \frac{q^{2}}{2}) (6 y + \frac{q^{2}}{2})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$6 y (6 y + \frac{q^{2}}{2}) - \frac{q^{2}}{2} (6 y + \frac{q^{2}}{2})$

Step 2.2

Apply the distributive property.

$6 y (6 y) + 6 y \frac{q^{2}}{2} - \frac{q^{2}}{2} (6 y + \frac{q^{2}}{2})$

Step 2.3

Apply the distributive property.

$6 y (6 y) + 6 y \frac{q^{2}}{2} - \frac{q^{2}}{2} (6 y) - \frac{q^{2}}{2} \cdot \frac{q^{2}}{2}$

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Rewrite using the commutative property of multiplication.

$6 \cdot 6 y \cdot y + 6 y \frac{q^{2}}{2} - \frac{q^{2}}{2} (6 y) - \frac{q^{2}}{2} \cdot \frac{q^{2}}{2}$

Step 3.1.2

Multiply $y$ by $y$ by adding the exponents.

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Step 3.1.2.1

Move $y$ .

$6 \cdot 6 (y \cdot y) + 6 y \frac{q^{2}}{2} - \frac{q^{2}}{2} (6 y) - \frac{q^{2}}{2} \cdot \frac{q^{2}}{2}$

Step 3.1.2.2

Multiply $y$ by $y$ .

$6 \cdot 6 y^{2} + 6 y \frac{q^{2}}{2} - \frac{q^{2}}{2} (6 y) - \frac{q^{2}}{2} \cdot \frac{q^{2}}{2}$

Step 3.1.3

Multiply $6$ by $6$ .

$36 y^{2} + 6 y \frac{q^{2}}{2} - \frac{q^{2}}{2} (6 y) - \frac{q^{2}}{2} \cdot \frac{q^{2}}{2}$

Step 3.1.4

Cancel the common factor of $2$ .

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Step 3.1.4.1

Factor $2$ out of $6 y$ .

$36 y^{2} + 2 (3 y) \frac{q^{2}}{2} - \frac{q^{2}}{2} (6 y) - \frac{q^{2}}{2} \cdot \frac{q^{2}}{2}$

Step 3.1.4.2

Cancel the common factor.

$36 y^{2} + 2 (3 y) \frac{q^{2}}{2} - \frac{q^{2}}{2} (6 y) - \frac{q^{2}}{2} \cdot \frac{q^{2}}{2}$

Step 3.1.4.3

Rewrite the expression.

$36 y^{2} + 3 y q^{2} - \frac{q^{2}}{2} (6 y) - \frac{q^{2}}{2} \cdot \frac{q^{2}}{2}$

Step 3.1.5

Cancel the common factor of $2$ .

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Step 3.1.5.1

Move the leading negative in $- \frac{q^{2}}{2}$ into the numerator.

$36 y^{2} + 3 y q^{2} + \frac{- q^{2}}{2} (6 y) - \frac{q^{2}}{2} \cdot \frac{q^{2}}{2}$

Step 3.1.5.2

Factor $2$ out of $6 y$ .

$36 y^{2} + 3 y q^{2} + \frac{- q^{2}}{2} (2 (3 y)) - \frac{q^{2}}{2} \cdot \frac{q^{2}}{2}$

Step 3.1.5.3

Cancel the common factor.

$36 y^{2} + 3 y q^{2} + \frac{- q^{2}}{2} (2 (3 y)) - \frac{q^{2}}{2} \cdot \frac{q^{2}}{2}$

Step 3.1.5.4

Rewrite the expression.

$36 y^{2} + 3 y q^{2} - q^{2} (3 y) - \frac{q^{2}}{2} \cdot \frac{q^{2}}{2}$

Step 3.1.6

Multiply $3$ by $- 1$ .

$36 y^{2} + 3 y q^{2} - 3 q^{2} y - \frac{q^{2}}{2} \cdot \frac{q^{2}}{2}$

Step 3.1.7

Multiply $- \frac{q^{2}}{2} \cdot \frac{q^{2}}{2}$ .

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Step 3.1.7.1

Multiply $\frac{q^{2}}{2}$ by $\frac{q^{2}}{2}$ .

$36 y^{2} + 3 y q^{2} - 3 q^{2} y - \frac{q^{2} q^{2}}{2 \cdot 2}$

Step 3.1.7.2

Multiply $q^{2}$ by $q^{2}$ by adding the exponents.

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Step 3.1.7.2.1

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

$36 y^{2} + 3 y q^{2} - 3 q^{2} y - \frac{q^{2 + 2}}{2 \cdot 2}$

Step 3.1.7.2.2

Add $2$ and $2$ .

$36 y^{2} + 3 y q^{2} - 3 q^{2} y - \frac{q^{4}}{2 \cdot 2}$

Step 3.1.7.3

Multiply $2$ by $2$ .

$36 y^{2} + 3 y q^{2} - 3 q^{2} y - \frac{q^{4}}{4}$

Step 3.2

Subtract $3 q^{2} y$ from $3 y q^{2}$ .

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Step 3.2.1

Move $y$ .

$36 y^{2} + 3 q^{2} y - 3 q^{2} y - \frac{q^{4}}{4}$

Step 3.2.2

Subtract $3 q^{2} y$ from $3 q^{2} y$ .

$36 y^{2} + 0 - \frac{q^{4}}{4}$

Step 3.3

Add $36 y^{2}$ and $0$ .

$36 y^{2} - \frac{q^{4}}{4}$