Finite Math Examples

${(x + 8)}^{2} - 80 = 0$

Step 1

Rewrite ${(x + 8)}^{2}$ as $(x + 8) (x + 8)$ .

$(x + 8) (x + 8) - 80 = 0$

Step 2

Expand $(x + 8) (x + 8)$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$x (x + 8) + 8 (x + 8) - 80 = 0$

Step 2.2

Apply the distributive property.

$x \cdot x + x \cdot 8 + 8 (x + 8) - 80 = 0$

Step 2.3

Apply the distributive property.

$x \cdot x + x \cdot 8 + 8 x + 8 \cdot 8 - 80 = 0$

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

Multiply $x$ by $x$ .

$x^{2} + x \cdot 8 + 8 x + 8 \cdot 8 - 80 = 0$

Step 3.1.2

Move $8$ to the left of $x$ .

$x^{2} + 8 x + 8 x + 8 \cdot 8 - 80 = 0$

Step 3.1.3

Multiply $8$ by $8$ .

$x^{2} + 8 x + 8 x + 64 - 80 = 0$

Step 3.2

Add $8 x$ and $8 x$ .

$x^{2} + 16 x + 64 - 80 = 0$

Step 4

Subtract $80$ from $64$ .

$x^{2} + 16 x - 16 = 0$

Step 5

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 6

Substitute the values $a = 1$ , $b = 16$ , and $c = - 16$ into the quadratic formula and solve for $x$ .

$\frac{- 16 \pm \sqrt{16^{2} - 4 \cdot (1 \cdot - 16)}}{2 \cdot 1}$

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise $16$ to the power of $2$ .

$x = \frac{- 16 \pm \sqrt{256 - 4 \cdot 1 \cdot - 16}}{2 \cdot 1}$

Step 7.1.2

Multiply $- 4 \cdot 1 \cdot - 16$ .

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Step 7.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 16 \pm \sqrt{256 - 4 \cdot - 16}}{2 \cdot 1}$

Step 7.1.2.2

Multiply $- 4$ by $- 16$ .

$x = \frac{- 16 \pm \sqrt{256 + 64}}{2 \cdot 1}$

Step 7.1.3

Add $256$ and $64$ .

$x = \frac{- 16 \pm \sqrt{320}}{2 \cdot 1}$

Step 7.1.4

Rewrite $320$ as $8^{2} \cdot 5$ .

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Step 7.1.4.1

Factor $64$ out of $320$ .

$x = \frac{- 16 \pm \sqrt{64 (5)}}{2 \cdot 1}$

Step 7.1.4.2

Rewrite $64$ as $8^{2}$ .

$x = \frac{- 16 \pm \sqrt{8^{2} \cdot 5}}{2 \cdot 1}$

Step 7.1.5

Pull terms out from under the radical.

$x = \frac{- 16 \pm 8 \sqrt{5}}{2 \cdot 1}$

Step 7.2

Multiply $2$ by $1$ .

$x = \frac{- 16 \pm 8 \sqrt{5}}{2}$

Step 7.3

Simplify $\frac{- 16 \pm 8 \sqrt{5}}{2}$ .

$x = - 8 \pm 4 \sqrt{5}$

Step 8

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Raise $16$ to the power of $2$ .

$x = \frac{- 16 \pm \sqrt{256 - 4 \cdot 1 \cdot - 16}}{2 \cdot 1}$

Step 8.1.2

Multiply $- 4 \cdot 1 \cdot - 16$ .

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Step 8.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 16 \pm \sqrt{256 - 4 \cdot - 16}}{2 \cdot 1}$

Step 8.1.2.2

Multiply $- 4$ by $- 16$ .

$x = \frac{- 16 \pm \sqrt{256 + 64}}{2 \cdot 1}$

Step 8.1.3

Add $256$ and $64$ .

$x = \frac{- 16 \pm \sqrt{320}}{2 \cdot 1}$

Step 8.1.4

Rewrite $320$ as $8^{2} \cdot 5$ .

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Step 8.1.4.1

Factor $64$ out of $320$ .

$x = \frac{- 16 \pm \sqrt{64 (5)}}{2 \cdot 1}$

Step 8.1.4.2

Rewrite $64$ as $8^{2}$ .

$x = \frac{- 16 \pm \sqrt{8^{2} \cdot 5}}{2 \cdot 1}$

Step 8.1.5

Pull terms out from under the radical.

$x = \frac{- 16 \pm 8 \sqrt{5}}{2 \cdot 1}$

Step 8.2

Multiply $2$ by $1$ .

$x = \frac{- 16 \pm 8 \sqrt{5}}{2}$

Step 8.3

Simplify $\frac{- 16 \pm 8 \sqrt{5}}{2}$ .

$x = - 8 \pm 4 \sqrt{5}$

Step 8.4

Change the $\pm$ to $+$ .

$x = - 8 + 4 \sqrt{5}$

Step 9

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 9.1

Simplify the numerator.

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Step 9.1.1

Raise $16$ to the power of $2$ .

$x = \frac{- 16 \pm \sqrt{256 - 4 \cdot 1 \cdot - 16}}{2 \cdot 1}$

Step 9.1.2

Multiply $- 4 \cdot 1 \cdot - 16$ .

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Step 9.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 16 \pm \sqrt{256 - 4 \cdot - 16}}{2 \cdot 1}$

Step 9.1.2.2

Multiply $- 4$ by $- 16$ .

$x = \frac{- 16 \pm \sqrt{256 + 64}}{2 \cdot 1}$

Step 9.1.3

Add $256$ and $64$ .

$x = \frac{- 16 \pm \sqrt{320}}{2 \cdot 1}$

Step 9.1.4

Rewrite $320$ as $8^{2} \cdot 5$ .

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Step 9.1.4.1

Factor $64$ out of $320$ .

$x = \frac{- 16 \pm \sqrt{64 (5)}}{2 \cdot 1}$

Step 9.1.4.2

Rewrite $64$ as $8^{2}$ .

$x = \frac{- 16 \pm \sqrt{8^{2} \cdot 5}}{2 \cdot 1}$

Step 9.1.5

Pull terms out from under the radical.

$x = \frac{- 16 \pm 8 \sqrt{5}}{2 \cdot 1}$

Step 9.2

Multiply $2$ by $1$ .

$x = \frac{- 16 \pm 8 \sqrt{5}}{2}$

Step 9.3

Simplify $\frac{- 16 \pm 8 \sqrt{5}}{2}$ .

$x = - 8 \pm 4 \sqrt{5}$

Step 9.4

Change the $\pm$ to $-$ .

$x = - 8 - 4 \sqrt{5}$

Step 10

The final answer is the combination of both solutions.

$x = - 8 + 4 \sqrt{5}, - 8 - 4 \sqrt{5}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$x = - 8 + 4 \sqrt{5}, - 8 - 4 \sqrt{5}$

Decimal Form:

$x = 0.94427190 \dots, - 16.94427190 \dots$