Algebra Examples

x^{2} = (x + 4) (x - 4) + 16

$x^{2} = (x + 4) (x - 4) + 16$

Step 1

Since

x

$x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

(x + 4) (x - 4) + 16 = x^{2}

$(x + 4) (x - 4) + 16 = x^{2}$

Step 2

Simplify

(x + 4) (x - 4) + 16

$(x + 4) (x - 4) + 16$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Expand

(x + 4) (x - 4)

$(x + 4) (x - 4)$ using the FOIL Method.

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Step 2.1.1.1

Apply the distributive property.

x (x - 4) + 4 (x - 4) + 16 = x^{2}

$x (x - 4) + 4 (x - 4) + 16 = x^{2}$

Step 2.1.1.2

Apply the distributive property.

x \cdot x + x \cdot - 4 + 4 (x - 4) + 16 = x^{2}

$x \cdot x + x \cdot - 4 + 4 (x - 4) + 16 = x^{2}$

Step 2.1.1.3

Apply the distributive property.

x \cdot x + x \cdot - 4 + 4 x + 4 \cdot - 4 + 16 = x^{2}

$x \cdot x + x \cdot - 4 + 4 x + 4 \cdot - 4 + 16 = x^{2}$

x \cdot x + x \cdot - 4 + 4 x + 4 \cdot - 4 + 16 = x^{2}

$x \cdot x + x \cdot - 4 + 4 x + 4 \cdot - 4 + 16 = x^{2}$

Step 2.1.2

Combine the opposite terms in

x \cdot x + x \cdot - 4 + 4 x + 4 \cdot - 4

$x \cdot x + x \cdot - 4 + 4 x + 4 \cdot - 4$ .

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Step 2.1.2.1

Reorder the factors in the terms

x \cdot - 4

$x \cdot - 4$ and

4 x

$4 x$ .

x \cdot x - 4 x + 4 x + 4 \cdot - 4 + 16 = x^{2}

$x \cdot x - 4 x + 4 x + 4 \cdot - 4 + 16 = x^{2}$

Step 2.1.2.2

Add

- 4 x

$- 4 x$ and

4 x

$4 x$ .

x \cdot x + 0 + 4 \cdot - 4 + 16 = x^{2}

$x \cdot x + 0 + 4 \cdot - 4 + 16 = x^{2}$

Step 2.1.2.3

Add

x \cdot x

$x \cdot x$ and

0

$0$ .

x \cdot x + 4 \cdot - 4 + 16 = x^{2}

$x \cdot x + 4 \cdot - 4 + 16 = x^{2}$

x \cdot x + 4 \cdot - 4 + 16 = x^{2}

$x \cdot x + 4 \cdot - 4 + 16 = x^{2}$

Step 2.1.3

Simplify each term.

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Step 2.1.3.1

Multiply

x

$x$ by

x

$x$ .

x^{2} + 4 \cdot - 4 + 16 = x^{2}

$x^{2} + 4 \cdot - 4 + 16 = x^{2}$

Step 2.1.3.2

Multiply

4

$4$ by

- 4

$- 4$ .

x^{2} - 16 + 16 = x^{2}

$x^{2} - 16 + 16 = x^{2}$

x^{2} - 16 + 16 = x^{2}

$x^{2} - 16 + 16 = x^{2}$

x^{2} - 16 + 16 = x^{2}

$x^{2} - 16 + 16 = x^{2}$

Step 2.2

Combine the opposite terms in

x^{2} - 16 + 16

$x^{2} - 16 + 16$ .

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Step 2.2.1

Add

- 16

$- 16$ and

16

$16$ .

x^{2} + 0 = x^{2}

$x^{2} + 0 = x^{2}$

Step 2.2.2

Add

x^{2}

$x^{2}$ and

0

$0$ .

x^{2} = x^{2}

$x^{2} = x^{2}$

x^{2} = x^{2}

$x^{2} = x^{2}$

x^{2} = x^{2}

$x^{2} = x^{2}$

Step 3

Move all terms containing

x

$x$ to the left side of the equation.

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Step 3.1

Subtract

x^{2}

$x^{2}$ from both sides of the equation.

x^{2} - x^{2} = 0

$x^{2} - x^{2} = 0$

Step 3.2

Subtract

x^{2}

$x^{2}$ from

x^{2}

$x^{2}$ .

0 = 0

$0 = 0$

0 = 0

$0 = 0$

Step 4

Since

0 = 0

$0 = 0$ , the equation will always be true for any value of

x

$x$ .

All real numbers

Step 5

The result can be shown in multiple forms.

All real numbers

Interval Notation:

(- \infty, \infty)

$(- \infty, \infty)$