Algebra Examples

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

Step 1

Use the binomial expansion theorem to find each term. The binomial theorem states ${(a + b)}^{n} = \sum_{k = 0}^{n} n C k \cdot (a^{n - k} b^{k})$ .

$\sum_{k = 0}^{2} \frac{2!}{(2 - k)! k!} \cdot {(2 x)}^{2 - k} \cdot {(- \frac{1}{2})}^{k}$

Step 2

Expand the summation.

$\frac{2!}{(2 - 0)! 0!} \cdot {(2 x)}^{2 - 0} \cdot {(- \frac{1}{2})}^{0} + \frac{2!}{(2 - 1)! 1!} \cdot {(2 x)}^{2 - 1} \cdot {(- \frac{1}{2})}^{1} + \frac{2!}{(2 - 2)! 2!} \cdot {(2 x)}^{2 - 2} \cdot {(- \frac{1}{2})}^{2}$

Step 3

Simplify the exponents for each term of the expansion.

$1 \cdot {(2 x)}^{2} \cdot {(- \frac{1}{2})}^{0} + 2 \cdot {(2 x)}^{1} \cdot {(- \frac{1}{2})}^{1} + 1 \cdot {(2 x)}^{0} \cdot {(- \frac{1}{2})}^{2}$

Step 4

Simplify each term.

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Step 4.1

Multiply ${(2 x)}^{2}$ by $1$ .

${(2 x)}^{2} \cdot {(- \frac{1}{2})}^{0} + 2 \cdot {(2 x)}^{1} \cdot {(- \frac{1}{2})}^{1} + 1 \cdot {(2 x)}^{0} \cdot {(- \frac{1}{2})}^{2}$

Step 4.2

Apply the product rule to $2 x$ .

$2^{2} x^{2} \cdot {(- \frac{1}{2})}^{0} + 2 \cdot {(2 x)}^{1} \cdot {(- \frac{1}{2})}^{1} + 1 \cdot {(2 x)}^{0} \cdot {(- \frac{1}{2})}^{2}$

Step 4.3

Raise $2$ to the power of $2$ .

$4 x^{2} \cdot {(- \frac{1}{2})}^{0} + 2 \cdot {(2 x)}^{1} \cdot {(- \frac{1}{2})}^{1} + 1 \cdot {(2 x)}^{0} \cdot {(- \frac{1}{2})}^{2}$

Step 4.4

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.4.1

Apply the product rule to $- \frac{1}{2}$ .

$4 x^{2} \cdot ({(- 1)}^{0} {(\frac{1}{2})}^{0}) + 2 \cdot {(2 x)}^{1} \cdot {(- \frac{1}{2})}^{1} + 1 \cdot {(2 x)}^{0} \cdot {(- \frac{1}{2})}^{2}$

Step 4.4.2

Apply the product rule to $\frac{1}{2}$ .

$4 x^{2} \cdot ({(- 1)}^{0} \frac{1^{0}}{2^{0}}) + 2 \cdot {(2 x)}^{1} \cdot {(- \frac{1}{2})}^{1} + 1 \cdot {(2 x)}^{0} \cdot {(- \frac{1}{2})}^{2}$

Step 4.5

Anything raised to $0$ is $1$ .

$4 x^{2} \cdot (1 \frac{1^{0}}{2^{0}}) + 2 \cdot {(2 x)}^{1} \cdot {(- \frac{1}{2})}^{1} + 1 \cdot {(2 x)}^{0} \cdot {(- \frac{1}{2})}^{2}$

Step 4.6

Multiply $\frac{1^{0}}{2^{0}}$ by $1$ .

$4 x^{2} \cdot \frac{1^{0}}{2^{0}} + 2 \cdot {(2 x)}^{1} \cdot {(- \frac{1}{2})}^{1} + 1 \cdot {(2 x)}^{0} \cdot {(- \frac{1}{2})}^{2}$

Step 4.7

Anything raised to $0$ is $1$ .

$4 x^{2} \cdot \frac{1}{2^{0}} + 2 \cdot {(2 x)}^{1} \cdot {(- \frac{1}{2})}^{1} + 1 \cdot {(2 x)}^{0} \cdot {(- \frac{1}{2})}^{2}$

Step 4.8

Anything raised to $0$ is $1$ .

$4 x^{2} \cdot \frac{1}{1} + 2 \cdot {(2 x)}^{1} \cdot {(- \frac{1}{2})}^{1} + 1 \cdot {(2 x)}^{0} \cdot {(- \frac{1}{2})}^{2}$

Step 4.9

Divide $1$ by $1$ .

$4 x^{2} \cdot 1 + 2 \cdot {(2 x)}^{1} \cdot {(- \frac{1}{2})}^{1} + 1 \cdot {(2 x)}^{0} \cdot {(- \frac{1}{2})}^{2}$

Step 4.10

Multiply $4$ by $1$ .

$4 x^{2} + 2 \cdot {(2 x)}^{1} \cdot {(- \frac{1}{2})}^{1} + 1 \cdot {(2 x)}^{0} \cdot {(- \frac{1}{2})}^{2}$

Step 4.11

Simplify.

$4 x^{2} + 2 \cdot (2 x) \cdot {(- \frac{1}{2})}^{1} + 1 \cdot {(2 x)}^{0} \cdot {(- \frac{1}{2})}^{2}$

Step 4.12

Multiply $2$ by $2$ .

$4 x^{2} + 4 x \cdot {(- \frac{1}{2})}^{1} + 1 \cdot {(2 x)}^{0} \cdot {(- \frac{1}{2})}^{2}$

Step 4.13

Simplify.

$4 x^{2} + 4 x \cdot (- \frac{1}{2}) + 1 \cdot {(2 x)}^{0} \cdot {(- \frac{1}{2})}^{2}$

Step 4.14

Cancel the common factor of $2$ .

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Step 4.14.1

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

$4 x^{2} + 4 x \cdot \frac{- 1}{2} + 1 \cdot {(2 x)}^{0} \cdot {(- \frac{1}{2})}^{2}$

Step 4.14.2

Factor $2$ out of $4 x$ .

$4 x^{2} + 2 (2 x) \cdot \frac{- 1}{2} + 1 \cdot {(2 x)}^{0} \cdot {(- \frac{1}{2})}^{2}$

Step 4.14.3

Cancel the common factor.

$4 x^{2} + 2 (2 x) \cdot \frac{- 1}{2} + 1 \cdot {(2 x)}^{0} \cdot {(- \frac{1}{2})}^{2}$

Step 4.14.4

Rewrite the expression.

$4 x^{2} + 2 x \cdot - 1 + 1 \cdot {(2 x)}^{0} \cdot {(- \frac{1}{2})}^{2}$

Step 4.15

Multiply $- 1$ by $2$ .

$4 x^{2} - 2 x + 1 \cdot {(2 x)}^{0} \cdot {(- \frac{1}{2})}^{2}$

Step 4.16

Multiply ${(2 x)}^{0}$ by $1$ .

$4 x^{2} - 2 x + {(2 x)}^{0} \cdot {(- \frac{1}{2})}^{2}$

Step 4.17

Apply the product rule to $2 x$ .

$4 x^{2} - 2 x + 2^{0} x^{0} \cdot {(- \frac{1}{2})}^{2}$

Step 4.18

Anything raised to $0$ is $1$ .

$4 x^{2} - 2 x + 1 x^{0} \cdot {(- \frac{1}{2})}^{2}$

Step 4.19

Multiply $x^{0}$ by $1$ .

$4 x^{2} - 2 x + x^{0} \cdot {(- \frac{1}{2})}^{2}$

Step 4.20

Anything raised to $0$ is $1$ .

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

Step 4.21

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

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

Step 4.22

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.22.1

Apply the product rule to $- \frac{1}{2}$ .

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

Step 4.22.2

Apply the product rule to $\frac{1}{2}$ .

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

Step 4.23

Raise $- 1$ to the power of $2$ .

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

Step 4.24

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

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

Step 4.25

One to any power is one.

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

Step 4.26

Raise $2$ to the power of $2$ .

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