Algebra Examples

${(3 d + \frac{1}{2} f)}^{2}$

Step 1

Combine fractions.

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Step 1.1

Combine $\frac{1}{2}$ and $f$ .

${(3 d + \frac{f}{2})}^{2}$

Step 1.2

Rewrite ${(3 d + \frac{f}{2})}^{2}$ as $(3 d + \frac{f}{2}) (3 d + \frac{f}{2})$ .

$(3 d + \frac{f}{2}) (3 d + \frac{f}{2})$

Step 2

Expand $(3 d + \frac{f}{2}) (3 d + \frac{f}{2})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$3 d (3 d + \frac{f}{2}) + \frac{f}{2} (3 d + \frac{f}{2})$

Step 2.2

Apply the distributive property.

$3 d (3 d) + 3 d \frac{f}{2} + \frac{f}{2} (3 d + \frac{f}{2})$

Step 2.3

Apply the distributive property.

$3 d (3 d) + 3 d \frac{f}{2} + \frac{f}{2} (3 d) + \frac{f}{2} \cdot \frac{f}{2}$

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.

$3 \cdot 3 d \cdot d + 3 d \frac{f}{2} + \frac{f}{2} (3 d) + \frac{f}{2} \cdot \frac{f}{2}$

Step 3.1.2

Multiply $d$ by $d$ by adding the exponents.

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Step 3.1.2.1

Move $d$ .

$3 \cdot 3 (d \cdot d) + 3 d \frac{f}{2} + \frac{f}{2} (3 d) + \frac{f}{2} \cdot \frac{f}{2}$

Step 3.1.2.2

Multiply $d$ by $d$ .

$3 \cdot 3 d^{2} + 3 d \frac{f}{2} + \frac{f}{2} (3 d) + \frac{f}{2} \cdot \frac{f}{2}$

Step 3.1.3

Multiply $3$ by $3$ .

$9 d^{2} + 3 d \frac{f}{2} + \frac{f}{2} (3 d) + \frac{f}{2} \cdot \frac{f}{2}$

Step 3.1.4

Multiply $3 d \frac{f}{2}$ .

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Step 3.1.4.1

Combine $3$ and $\frac{f}{2}$ .

$9 d^{2} + d \frac{3 f}{2} + \frac{f}{2} (3 d) + \frac{f}{2} \cdot \frac{f}{2}$

Step 3.1.4.2

Combine $d$ and $\frac{3 f}{2}$ .

$9 d^{2} + \frac{d (3 f)}{2} + \frac{f}{2} (3 d) + \frac{f}{2} \cdot \frac{f}{2}$

Step 3.1.5

Move $3$ to the left of $d$ .

$9 d^{2} + \frac{3 \cdot d f}{2} + \frac{f}{2} (3 d) + \frac{f}{2} \cdot \frac{f}{2}$

Step 3.1.6

Rewrite using the commutative property of multiplication.

$9 d^{2} + \frac{3 d f}{2} + 3 \frac{f}{2} d + \frac{f}{2} \cdot \frac{f}{2}$

Step 3.1.7

Combine $3$ and $\frac{f}{2}$ .

$9 d^{2} + \frac{3 d f}{2} + \frac{3 f}{2} d + \frac{f}{2} \cdot \frac{f}{2}$

Step 3.1.8

Combine $\frac{3 f}{2}$ and $d$ .

$9 d^{2} + \frac{3 d f}{2} + \frac{3 f d}{2} + \frac{f}{2} \cdot \frac{f}{2}$

Step 3.1.9

Multiply $\frac{f}{2} \cdot \frac{f}{2}$ .

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Step 3.1.9.1

Multiply $\frac{f}{2}$ by $\frac{f}{2}$ .

$9 d^{2} + \frac{3 d f}{2} + \frac{3 f d}{2} + \frac{f \cdot f}{2 \cdot 2}$

Step 3.1.9.2

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

$9 d^{2} + \frac{3 d f}{2} + \frac{3 f d}{2} + \frac{f^{1} f}{2 \cdot 2}$

Step 3.1.9.3

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

$9 d^{2} + \frac{3 d f}{2} + \frac{3 f d}{2} + \frac{f^{1} f^{1}}{2 \cdot 2}$

Step 3.1.9.4

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

$9 d^{2} + \frac{3 d f}{2} + \frac{3 f d}{2} + \frac{f^{1 + 1}}{2 \cdot 2}$

Step 3.1.9.5

Add $1$ and $1$ .

$9 d^{2} + \frac{3 d f}{2} + \frac{3 f d}{2} + \frac{f^{2}}{2 \cdot 2}$

Step 3.1.9.6

Multiply $2$ by $2$ .

$9 d^{2} + \frac{3 d f}{2} + \frac{3 f d}{2} + \frac{f^{2}}{4}$

Step 3.2

Add $\frac{3 d f}{2}$ and $\frac{3 f d}{2}$ .

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Step 3.2.1

Move $f$ .

$9 d^{2} + \frac{3 d f}{2} + \frac{3 d f}{2} + \frac{f^{2}}{4}$

Step 3.2.2

Add $\frac{3 d f}{2}$ and $\frac{3 d f}{2}$ .

$9 d^{2} + 2 \frac{3 d f}{2} + \frac{f^{2}}{4}$

Step 4

Cancel the common factor of $2$ .

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Step 4.1

Cancel the common factor.

$9 d^{2} + 2 \frac{3 d f}{2} + \frac{f^{2}}{4}$

Step 4.2

Rewrite the expression.

$9 d^{2} + 3 d f + \frac{f^{2}}{4}$