Algebra Examples

$f (x) = 8 x^{2} - 16 x - 15$

Step 1

Set $8 x^{2} - 16 x - 15$ equal to $0$ .

$8 x^{2} - 16 x - 15 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Use the quadratic formula to find the solutions.

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

Step 2.2

Substitute the values $a = 8$ , $b = - 16$ , and $c = - 15$ into the quadratic formula and solve for $x$ .

$\frac{16 \pm \sqrt{{(- 16)}^{2} - 4 \cdot (8 \cdot - 15)}}{2 \cdot 8}$

Step 2.3

Simplify.

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Step 2.3.1

Simplify the numerator.

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Step 2.3.1.1

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

$x = \frac{16 \pm \sqrt{256 - 4 \cdot 8 \cdot - 15}}{2 \cdot 8}$

Step 2.3.1.2

Multiply $- 4 \cdot 8 \cdot - 15$ .

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Step 2.3.1.2.1

Multiply $- 4$ by $8$ .

$x = \frac{16 \pm \sqrt{256 - 32 \cdot - 15}}{2 \cdot 8}$

Step 2.3.1.2.2

Multiply $- 32$ by $- 15$ .

$x = \frac{16 \pm \sqrt{256 + 480}}{2 \cdot 8}$

Step 2.3.1.3

Add $256$ and $480$ .

$x = \frac{16 \pm \sqrt{736}}{2 \cdot 8}$

Step 2.3.1.4

Rewrite $736$ as $4^{2} \cdot 46$ .

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Step 2.3.1.4.1

Factor $16$ out of $736$ .

$x = \frac{16 \pm \sqrt{16 (46)}}{2 \cdot 8}$

Step 2.3.1.4.2

Rewrite $16$ as $4^{2}$ .

$x = \frac{16 \pm \sqrt{4^{2} \cdot 46}}{2 \cdot 8}$

Step 2.3.1.5

Pull terms out from under the radical.

$x = \frac{16 \pm 4 \sqrt{46}}{2 \cdot 8}$

Step 2.3.2

Multiply $2$ by $8$ .

$x = \frac{16 \pm 4 \sqrt{46}}{16}$

Step 2.3.3

Simplify $\frac{16 \pm 4 \sqrt{46}}{16}$ .

$x = \frac{4 \pm \sqrt{46}}{4}$

Step 2.4

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

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Step 2.4.1

Simplify the numerator.

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Step 2.4.1.1

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

$x = \frac{16 \pm \sqrt{256 - 4 \cdot 8 \cdot - 15}}{2 \cdot 8}$

Step 2.4.1.2

Multiply $- 4 \cdot 8 \cdot - 15$ .

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Step 2.4.1.2.1

Multiply $- 4$ by $8$ .

$x = \frac{16 \pm \sqrt{256 - 32 \cdot - 15}}{2 \cdot 8}$

Step 2.4.1.2.2

Multiply $- 32$ by $- 15$ .

$x = \frac{16 \pm \sqrt{256 + 480}}{2 \cdot 8}$

Step 2.4.1.3

Add $256$ and $480$ .

$x = \frac{16 \pm \sqrt{736}}{2 \cdot 8}$

Step 2.4.1.4

Rewrite $736$ as $4^{2} \cdot 46$ .

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Step 2.4.1.4.1

Factor $16$ out of $736$ .

$x = \frac{16 \pm \sqrt{16 (46)}}{2 \cdot 8}$

Step 2.4.1.4.2

Rewrite $16$ as $4^{2}$ .

$x = \frac{16 \pm \sqrt{4^{2} \cdot 46}}{2 \cdot 8}$

Step 2.4.1.5

Pull terms out from under the radical.

$x = \frac{16 \pm 4 \sqrt{46}}{2 \cdot 8}$

Step 2.4.2

Multiply $2$ by $8$ .

$x = \frac{16 \pm 4 \sqrt{46}}{16}$

Step 2.4.3

Simplify $\frac{16 \pm 4 \sqrt{46}}{16}$ .

$x = \frac{4 \pm \sqrt{46}}{4}$

Step 2.4.4

Change the $\pm$ to $+$ .

$x = \frac{4 + \sqrt{46}}{4}$

Step 2.5

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

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Step 2.5.1

Simplify the numerator.

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Step 2.5.1.1

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

$x = \frac{16 \pm \sqrt{256 - 4 \cdot 8 \cdot - 15}}{2 \cdot 8}$

Step 2.5.1.2

Multiply $- 4 \cdot 8 \cdot - 15$ .

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Step 2.5.1.2.1

Multiply $- 4$ by $8$ .

$x = \frac{16 \pm \sqrt{256 - 32 \cdot - 15}}{2 \cdot 8}$

Step 2.5.1.2.2

Multiply $- 32$ by $- 15$ .

$x = \frac{16 \pm \sqrt{256 + 480}}{2 \cdot 8}$

Step 2.5.1.3

Add $256$ and $480$ .

$x = \frac{16 \pm \sqrt{736}}{2 \cdot 8}$

Step 2.5.1.4

Rewrite $736$ as $4^{2} \cdot 46$ .

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Step 2.5.1.4.1

Factor $16$ out of $736$ .

$x = \frac{16 \pm \sqrt{16 (46)}}{2 \cdot 8}$

Step 2.5.1.4.2

Rewrite $16$ as $4^{2}$ .

$x = \frac{16 \pm \sqrt{4^{2} \cdot 46}}{2 \cdot 8}$

Step 2.5.1.5

Pull terms out from under the radical.

$x = \frac{16 \pm 4 \sqrt{46}}{2 \cdot 8}$

Step 2.5.2

Multiply $2$ by $8$ .

$x = \frac{16 \pm 4 \sqrt{46}}{16}$

Step 2.5.3

Simplify $\frac{16 \pm 4 \sqrt{46}}{16}$ .

$x = \frac{4 \pm \sqrt{46}}{4}$

Step 2.5.4

Change the $\pm$ to $-$ .

$x = \frac{4 - \sqrt{46}}{4}$

Step 2.6

The final answer is the combination of both solutions.

$x = \frac{4 + \sqrt{46}}{4}$ (Multiplicity of $1$ )

$x = \frac{4 - \sqrt{46}}{4}$ (Multiplicity of $1$ )

$x = \frac{4 + \sqrt{46}}{4}$ (Multiplicity of $1$ )

$x = \frac{4 - \sqrt{46}}{4}$ (Multiplicity of $1$ )

Step 3