Pre-Algebra Examples

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

Step 1

Rewrite the equation as $- 16 x^{2} + 64 x + 16 = 0$ .

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

Step 2

Factor $- 16$ out of $- 16 x^{2} + 64 x + 16$ .

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Step 2.1

Factor $- 16$ out of $- 16 x^{2}$ .

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

Step 2.2

Factor $- 16$ out of $64 x$ .

$- 16 x^{2} - 16 (- 4 x) + 16 = 0$

Step 2.3

Factor $- 16$ out of $16$ .

$- 16 x^{2} - 16 (- 4 x) - 16 \cdot - 1 = 0$

Step 2.4

Factor $- 16$ out of $- 16 (x^{2}) - 16 (- 4 x)$ .

$- 16 (x^{2} - 4 x) - 16 \cdot - 1 = 0$

Step 2.5

Factor $- 16$ out of $- 16 (x^{2} - 4 x) - 16 (- 1)$ .

$- 16 (x^{2} - 4 x - 1) = 0$

Step 3

Divide each term in $- 16 (x^{2} - 4 x - 1) = 0$ by $- 16$ and simplify.

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Step 3.1

Divide each term in $- 16 (x^{2} - 4 x - 1) = 0$ by $- 16$ .

$\frac{- 16 (x^{2} - 4 x - 1)}{- 16} = \frac{0}{- 16}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $- 16$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{- 16 (x^{2} - 4 x - 1)}{- 16} = \frac{0}{- 16}$

Step 3.2.1.2

Divide $x^{2} - 4 x - 1$ by $1$ .

$x^{2} - 4 x - 1 = \frac{0}{- 16}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $0$ by $- 16$ .

$x^{2} - 4 x - 1 = 0$

Step 4

Use the quadratic formula to find the solutions.

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

Step 5

Substitute the values $a = 1$ , $b = - 4$ , and $c = - 1$ into the quadratic formula and solve for $x$ .

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

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $- 4$ to the power of $2$ .

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

Step 6.1.2

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

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Step 6.1.2.1

Multiply $- 4$ by $1$ .

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

Step 6.1.2.2

Multiply $- 4$ by $- 1$ .

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

Step 6.1.3

Add $16$ and $4$ .

$x = \frac{4 \pm \sqrt{20}}{2 \cdot 1}$

Step 6.1.4

Rewrite $20$ as $2^{2} \cdot 5$ .

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Step 6.1.4.1

Factor $4$ out of $20$ .

$x = \frac{4 \pm \sqrt{4 (5)}}{2 \cdot 1}$

Step 6.1.4.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{4 \pm \sqrt{2^{2} \cdot 5}}{2 \cdot 1}$

Step 6.1.5

Pull terms out from under the radical.

$x = \frac{4 \pm 2 \sqrt{5}}{2 \cdot 1}$

Step 6.2

Multiply $2$ by $1$ .

$x = \frac{4 \pm 2 \sqrt{5}}{2}$

Step 6.3

Simplify $\frac{4 \pm 2 \sqrt{5}}{2}$ .

$x = 2 \pm \sqrt{5}$

Step 7

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

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise $- 4$ to the power of $2$ .

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

Step 7.1.2

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

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Step 7.1.2.1

Multiply $- 4$ by $1$ .

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

Step 7.1.2.2

Multiply $- 4$ by $- 1$ .

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

Step 7.1.3

Add $16$ and $4$ .

$x = \frac{4 \pm \sqrt{20}}{2 \cdot 1}$

Step 7.1.4

Rewrite $20$ as $2^{2} \cdot 5$ .

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Step 7.1.4.1

Factor $4$ out of $20$ .

$x = \frac{4 \pm \sqrt{4 (5)}}{2 \cdot 1}$

Step 7.1.4.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{4 \pm \sqrt{2^{2} \cdot 5}}{2 \cdot 1}$

Step 7.1.5

Pull terms out from under the radical.

$x = \frac{4 \pm 2 \sqrt{5}}{2 \cdot 1}$

Step 7.2

Multiply $2$ by $1$ .

$x = \frac{4 \pm 2 \sqrt{5}}{2}$

Step 7.3

Simplify $\frac{4 \pm 2 \sqrt{5}}{2}$ .

$x = 2 \pm \sqrt{5}$

Step 7.4

Change the $\pm$ to $+$ .

$x = 2 + \sqrt{5}$

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 $- 4$ to the power of $2$ .

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

Step 8.1.2

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

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Step 8.1.2.1

Multiply $- 4$ by $1$ .

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

Step 8.1.2.2

Multiply $- 4$ by $- 1$ .

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

Step 8.1.3

Add $16$ and $4$ .

$x = \frac{4 \pm \sqrt{20}}{2 \cdot 1}$

Step 8.1.4

Rewrite $20$ as $2^{2} \cdot 5$ .

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Step 8.1.4.1

Factor $4$ out of $20$ .

$x = \frac{4 \pm \sqrt{4 (5)}}{2 \cdot 1}$

Step 8.1.4.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{4 \pm \sqrt{2^{2} \cdot 5}}{2 \cdot 1}$

Step 8.1.5

Pull terms out from under the radical.

$x = \frac{4 \pm 2 \sqrt{5}}{2 \cdot 1}$

Step 8.2

Multiply $2$ by $1$ .

$x = \frac{4 \pm 2 \sqrt{5}}{2}$

Step 8.3

Simplify $\frac{4 \pm 2 \sqrt{5}}{2}$ .

$x = 2 \pm \sqrt{5}$

Step 8.4

Change the $\pm$ to $-$ .

$x = 2 - \sqrt{5}$

Step 9

The final answer is the combination of both solutions.

$x = 2 + \sqrt{5}, 2 - \sqrt{5}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$x = 2 + \sqrt{5}, 2 - \sqrt{5}$

Decimal Form:

$x = 4.23606797 \dots, - 0.23606797 \dots$