Algebra Examples

{(2 x - 5)}^{2}

${(2 x - 5)}^{2}$

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})

${(a + b)}^{n} = \sum_{k = 0}^{n} n C k \cdot (a^{n - k} b^{k})$ .

\sum_{k = 0}^{2} \frac{2!}{(2 - k)! k!} \cdot {(2 x)}^{2 - k} \cdot {(- 5)}^{k}

$\sum_{k = 0}^{2} \frac{2!}{(2 - k)! k!} \cdot {(2 x)}^{2 - k} \cdot {(- 5)}^{k}$

Step 2

Expand the summation.

\frac{2!}{(2 - 0)! 0!} \cdot {(2 x)}^{2 - 0} \cdot {(- 5)}^{0} + \frac{2!}{(2 - 1)! 1!} \cdot {(2 x)}^{2 - 1} \cdot {(- 5)}^{1} + \frac{2!}{(2 - 2)! 2!} \cdot {(2 x)}^{2 - 2} \cdot {(- 5)}^{2}

$\frac{2!}{(2 - 0)! 0!} \cdot {(2 x)}^{2 - 0} \cdot {(- 5)}^{0} + \frac{2!}{(2 - 1)! 1!} \cdot {(2 x)}^{2 - 1} \cdot {(- 5)}^{1} + \frac{2!}{(2 - 2)! 2!} \cdot {(2 x)}^{2 - 2} \cdot {(- 5)}^{2}$

Step 3

Simplify the exponents for each term of the expansion.

1 \cdot {(2 x)}^{2} \cdot {(- 5)}^{0} + 2 \cdot {(2 x)}^{1} \cdot {(- 5)}^{1} + 1 \cdot {(2 x)}^{0} \cdot {(- 5)}^{2}

$1 \cdot {(2 x)}^{2} \cdot {(- 5)}^{0} + 2 \cdot {(2 x)}^{1} \cdot {(- 5)}^{1} + 1 \cdot {(2 x)}^{0} \cdot {(- 5)}^{2}$

Step 4

Simplify each term.

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Step 4.1

Multiply

{(2 x)}^{2}

${(2 x)}^{2}$ by

1

$1$ .

{(2 x)}^{2} \cdot {(- 5)}^{0} + 2 \cdot {(2 x)}^{1} \cdot {(- 5)}^{1} + 1 \cdot {(2 x)}^{0} \cdot {(- 5)}^{2}

${(2 x)}^{2} \cdot {(- 5)}^{0} + 2 \cdot {(2 x)}^{1} \cdot {(- 5)}^{1} + 1 \cdot {(2 x)}^{0} \cdot {(- 5)}^{2}$

Step 4.2

Apply the product rule to

2 x

$2 x$ .

2^{2} x^{2} \cdot {(- 5)}^{0} + 2 \cdot {(2 x)}^{1} \cdot {(- 5)}^{1} + 1 \cdot {(2 x)}^{0} \cdot {(- 5)}^{2}

$2^{2} x^{2} \cdot {(- 5)}^{0} + 2 \cdot {(2 x)}^{1} \cdot {(- 5)}^{1} + 1 \cdot {(2 x)}^{0} \cdot {(- 5)}^{2}$

Step 4.3

Raise

2

$2$ to the power of

2

$2$ .

4 x^{2} \cdot {(- 5)}^{0} + 2 \cdot {(2 x)}^{1} \cdot {(- 5)}^{1} + 1 \cdot {(2 x)}^{0} \cdot {(- 5)}^{2}

$4 x^{2} \cdot {(- 5)}^{0} + 2 \cdot {(2 x)}^{1} \cdot {(- 5)}^{1} + 1 \cdot {(2 x)}^{0} \cdot {(- 5)}^{2}$

Step 4.4

Anything raised to

0

$0$ is

1

$1$ .

4 x^{2} \cdot 1 + 2 \cdot {(2 x)}^{1} \cdot {(- 5)}^{1} + 1 \cdot {(2 x)}^{0} \cdot {(- 5)}^{2}

$4 x^{2} \cdot 1 + 2 \cdot {(2 x)}^{1} \cdot {(- 5)}^{1} + 1 \cdot {(2 x)}^{0} \cdot {(- 5)}^{2}$

Step 4.5

Multiply

4

$4$ by

1

$1$ .

4 x^{2} + 2 \cdot {(2 x)}^{1} \cdot {(- 5)}^{1} + 1 \cdot {(2 x)}^{0} \cdot {(- 5)}^{2}

$4 x^{2} + 2 \cdot {(2 x)}^{1} \cdot {(- 5)}^{1} + 1 \cdot {(2 x)}^{0} \cdot {(- 5)}^{2}$

Step 4.6

Simplify.

4 x^{2} + 2 \cdot (2 x) \cdot {(- 5)}^{1} + 1 \cdot {(2 x)}^{0} \cdot {(- 5)}^{2}

$4 x^{2} + 2 \cdot (2 x) \cdot {(- 5)}^{1} + 1 \cdot {(2 x)}^{0} \cdot {(- 5)}^{2}$

Step 4.7

Multiply

2

$2$ by

2

$2$ .

4 x^{2} + 4 x \cdot {(- 5)}^{1} + 1 \cdot {(2 x)}^{0} \cdot {(- 5)}^{2}

$4 x^{2} + 4 x \cdot {(- 5)}^{1} + 1 \cdot {(2 x)}^{0} \cdot {(- 5)}^{2}$

Step 4.8

Evaluate the exponent.

4 x^{2} + 4 x \cdot - 5 + 1 \cdot {(2 x)}^{0} \cdot {(- 5)}^{2}

$4 x^{2} + 4 x \cdot - 5 + 1 \cdot {(2 x)}^{0} \cdot {(- 5)}^{2}$

Step 4.9

Multiply

- 5

$- 5$ by

4

$4$ .

4 x^{2} - 20 x + 1 \cdot {(2 x)}^{0} \cdot {(- 5)}^{2}

$4 x^{2} - 20 x + 1 \cdot {(2 x)}^{0} \cdot {(- 5)}^{2}$

Step 4.10

Multiply

{(2 x)}^{0}

${(2 x)}^{0}$ by

1

$1$ .

4 x^{2} - 20 x + {(2 x)}^{0} \cdot {(- 5)}^{2}

$4 x^{2} - 20 x + {(2 x)}^{0} \cdot {(- 5)}^{2}$

Step 4.11

Apply the product rule to

2 x

$2 x$ .

4 x^{2} - 20 x + 2^{0} x^{0} \cdot {(- 5)}^{2}

$4 x^{2} - 20 x + 2^{0} x^{0} \cdot {(- 5)}^{2}$

Step 4.12

Anything raised to

0

$0$ is

1

$1$ .

4 x^{2} - 20 x + 1 x^{0} \cdot {(- 5)}^{2}

$4 x^{2} - 20 x + 1 x^{0} \cdot {(- 5)}^{2}$

Step 4.13

Multiply

x^{0}

$x^{0}$ by

1

$1$ .

4 x^{2} - 20 x + x^{0} \cdot {(- 5)}^{2}

$4 x^{2} - 20 x + x^{0} \cdot {(- 5)}^{2}$

Step 4.14

Anything raised to

0

$0$ is

1

$1$ .

4 x^{2} - 20 x + 1 \cdot {(- 5)}^{2}

$4 x^{2} - 20 x + 1 \cdot {(- 5)}^{2}$

Step 4.15

Multiply

{(- 5)}^{2}

${(- 5)}^{2}$ by

1

$1$ .

4 x^{2} - 20 x + {(- 5)}^{2}

$4 x^{2} - 20 x + {(- 5)}^{2}$

Step 4.16

Raise

- 5

$- 5$ to the power of

2

$2$ .

4 x^{2} - 20 x + 25

$4 x^{2} - 20 x + 25$

4 x^{2} - 20 x + 25

$4 x^{2} - 20 x + 25$