Precalculus Examples

$x^{2} - 14 x + 17 = 0$

Step 1

Use the quadratic formula to find the solutions.

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

Step 2

Substitute the values $a = 1$ , $b = - 14$ , and $c = 17$ into the quadratic formula and solve for $x$ .

$\frac{14 \pm \sqrt{{(- 14)}^{2} - 4 \cdot (1 \cdot 17)}}{2 \cdot 1}$

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Raise $- 14$ to the power of $2$ .

$x = \frac{14 \pm \sqrt{196 - 4 \cdot 1 \cdot 17}}{2 \cdot 1}$

Step 3.1.2

Multiply $- 4 \cdot 1 \cdot 17$ .

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Step 3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{14 \pm \sqrt{196 - 4 \cdot 17}}{2 \cdot 1}$

Step 3.1.2.2

Multiply $- 4$ by $17$ .

$x = \frac{14 \pm \sqrt{196 - 68}}{2 \cdot 1}$

Step 3.1.3

Subtract $68$ from $196$ .

$x = \frac{14 \pm \sqrt{128}}{2 \cdot 1}$

Step 3.1.4

Rewrite $128$ as $8^{2} \cdot 2$ .

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Step 3.1.4.1

Factor $64$ out of $128$ .

$x = \frac{14 \pm \sqrt{64 (2)}}{2 \cdot 1}$

Step 3.1.4.2

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

$x = \frac{14 \pm \sqrt{8^{2} \cdot 2}}{2 \cdot 1}$

Step 3.1.5

Pull terms out from under the radical.

$x = \frac{14 \pm 8 \sqrt{2}}{2 \cdot 1}$

Step 3.2

Multiply $2$ by $1$ .

$x = \frac{14 \pm 8 \sqrt{2}}{2}$

Step 3.3

Simplify $\frac{14 \pm 8 \sqrt{2}}{2}$ .

$x = 7 \pm 4 \sqrt{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = 7 \pm 4 \sqrt{2}$

Decimal Form:

$x = 12.65685424 \dots, 1.34314575 \dots$