Algebra Examples

${(2 x - \frac{1}{2})}^{2}$

Step 1

Rewrite ${(2 x - \frac{1}{2})}^{2}$ as $(2 x - \frac{1}{2}) (2 x - \frac{1}{2})$ .

$(2 x - \frac{1}{2}) (2 x - \frac{1}{2})$

Step 2

Expand $(2 x - \frac{1}{2}) (2 x - \frac{1}{2})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$2 x (2 x - \frac{1}{2}) - \frac{1}{2} (2 x - \frac{1}{2})$

Step 2.2

Apply the distributive property.

$2 x (2 x) + 2 x (- \frac{1}{2}) - \frac{1}{2} (2 x - \frac{1}{2})$

Step 2.3

Apply the distributive property.

$2 x (2 x) + 2 x (- \frac{1}{2}) - \frac{1}{2} (2 x) - \frac{1}{2} (- \frac{1}{2})$

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.

$2 \cdot 2 x \cdot x + 2 x (- \frac{1}{2}) - \frac{1}{2} (2 x) - \frac{1}{2} (- \frac{1}{2})$

Step 3.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 3.1.2.1

Move $x$ .

$2 \cdot 2 (x \cdot x) + 2 x (- \frac{1}{2}) - \frac{1}{2} (2 x) - \frac{1}{2} (- \frac{1}{2})$

Step 3.1.2.2

Multiply $x$ by $x$ .

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

Step 3.1.3

Multiply $2$ by $2$ .

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

Step 3.1.4

Cancel the common factor of $2$ .

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Step 3.1.4.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

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

Step 3.1.4.2

Factor $2$ out of $2 x$ .

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

Step 3.1.4.3

Cancel the common factor.

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

Step 3.1.4.4

Rewrite the expression.

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

Step 3.1.5

Move $- 1$ to the left of $x$ .

$4 x^{2} - 1 \cdot x - \frac{1}{2} (2 x) - \frac{1}{2} (- \frac{1}{2})$

Step 3.1.6

Rewrite $- 1 x$ as $- x$ .

$4 x^{2} - x - \frac{1}{2} (2 x) - \frac{1}{2} (- \frac{1}{2})$

Step 3.1.7

Cancel the common factor of $2$ .

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Step 3.1.7.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

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

Step 3.1.7.2

Factor $2$ out of $2 x$ .

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

Step 3.1.7.3

Cancel the common factor.

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

Step 3.1.7.4

Rewrite the expression.

$4 x^{2} - x - 1 x - \frac{1}{2} (- \frac{1}{2})$

Step 3.1.8

Rewrite $- 1 x$ as $- x$ .

$4 x^{2} - x - x - \frac{1}{2} (- \frac{1}{2})$

Step 3.1.9

Multiply $- \frac{1}{2} (- \frac{1}{2})$ .

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Step 3.1.9.1

Multiply $- 1$ by $- 1$ .

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

Step 3.1.9.2

Multiply $\frac{1}{2}$ by $1$ .

$4 x^{2} - x - x + \frac{1}{2} \cdot \frac{1}{2}$

Step 3.1.9.3

Multiply $\frac{1}{2}$ by $\frac{1}{2}$ .

$4 x^{2} - x - x + \frac{1}{2 \cdot 2}$

Step 3.1.9.4

Multiply $2$ by $2$ .

$4 x^{2} - x - x + \frac{1}{4}$

Step 3.2

Subtract $x$ from $- x$ .

$4 x^{2} - 2 x + \frac{1}{4}$