Solve Using the Quadratic Formula x^2-14x+67=0

$x^{2} - 14 x + 67 = 0$

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 $c = 67$ into the quadratic formula and solve for $x$ .

$\frac{14 \pm \sqrt{{(- 14)}^{2} - 4 \cdot (1 \cdot 67)}}{2 \cdot 1}$

Step 3

Simplify.

Simplify the numerator.

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

$x = \frac{14 \pm \sqrt{196 - 4 \cdot 1 \cdot 67}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 67$ .

Multiply $- 4$ by $1$ .

$x = \frac{14 \pm \sqrt{196 - 4 \cdot 67}}{2 \cdot 1}$

Multiply $- 4$ by $67$ .

$x = \frac{14 \pm \sqrt{196 - 268}}{2 \cdot 1}$

Subtract $268$ from $196$ .

$x = \frac{14 \pm \sqrt{- 72}}{2 \cdot 1}$

Rewrite $- 72$ as $- 1 (72)$ .

$x = \frac{14 \pm \sqrt{- 1 \cdot 72}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (72)}$ as $\sqrt{- 1} \cdot \sqrt{72}$ .

$x = \frac{14 \pm \sqrt{- 1} \cdot \sqrt{72}}{2 \cdot 1}$

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{14 \pm i \cdot \sqrt{72}}{2 \cdot 1}$

Rewrite $72$ as $6^{2} \cdot 2$ .

Factor $36$ out of $72$ .

$x = \frac{14 \pm i \cdot \sqrt{36 (2)}}{2 \cdot 1}$

Rewrite $36$ as $6^{2}$ .

$x = \frac{14 \pm i \cdot \sqrt{6^{2} \cdot 2}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{14 \pm i \cdot (6 \sqrt{2})}{2 \cdot 1}$

Move $6$ to the left of $i$ .

$x = \frac{14 \pm 6 i \sqrt{2}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{14 \pm 6 i \sqrt{2}}{2}$

Simplify $\frac{14 \pm 6 i \sqrt{2}}{2}$ .

$x = 7 \pm 3 i \sqrt{2}$