Solve Using the Quadratic Formula x^2-14x+47=0

$x^{2} - 14 x + 47 = 0$

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 $c = 47$ into the quadratic formula and solve for $x$ .

$\frac{14 \pm \sqrt{{(- 14)}^{2} - 4 \cdot (1 \cdot 47)}}{2 \cdot 1}$

Step 3

Simplify.

Simplify the numerator.

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

$x = \frac{14 \pm \sqrt{196 - 4 \cdot 1 \cdot 47}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 47$ .

Multiply $- 4$ by $1$ .

$x = \frac{14 \pm \sqrt{196 - 4 \cdot 47}}{2 \cdot 1}$

Multiply $- 4$ by $47$ .

$x = \frac{14 \pm \sqrt{196 - 188}}{2 \cdot 1}$

Subtract $188$ from $196$ .

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

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

Factor $4$ out of $8$ .

$x = \frac{14 \pm \sqrt{4 (2)}}{2 \cdot 1}$

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

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

Pull terms out from under the radical.

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

Multiply $2$ by $1$ .

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

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

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

Step 4

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 8.41421356 \dots, 5.58578643 \dots$