Solve for x x^2-12x+59=0

$x^{2} - 12 x + 59 = 0$

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 $a = 1$ , $b = - 12$ , and $c = 59$ into the quadratic formula and solve for $x$ .

$\frac{12 \pm \sqrt{{(- 12)}^{2} - 4 \cdot (1 \cdot 59)}}{2 \cdot 1}$

Step 3

Simplify.

Simplify the numerator.

Raise $- 12$ to the power of $2$ .

$x = \frac{12 \pm \sqrt{144 - 4 \cdot 1 \cdot 59}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 59$ .

Multiply $- 4$ by $1$ .

$x = \frac{12 \pm \sqrt{144 - 4 \cdot 59}}{2 \cdot 1}$

Multiply $- 4$ by $59$ .

$x = \frac{12 \pm \sqrt{144 - 236}}{2 \cdot 1}$

Subtract $236$ from $144$ .

$x = \frac{12 \pm \sqrt{- 92}}{2 \cdot 1}$

Rewrite $- 92$ as $- 1 (92)$ .

$x = \frac{12 \pm \sqrt{- 1 \cdot 92}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (92)}$ as $\sqrt{- 1} \cdot \sqrt{92}$ .

$x = \frac{12 \pm \sqrt{- 1} \cdot \sqrt{92}}{2 \cdot 1}$

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

$x = \frac{12 \pm i \cdot \sqrt{92}}{2 \cdot 1}$

Rewrite $92$ as $2^{2} \cdot 23$ .

Factor $4$ out of $92$ .

$x = \frac{12 \pm i \cdot \sqrt{4 (23)}}{2 \cdot 1}$

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

$x = \frac{12 \pm i \cdot \sqrt{2^{2} \cdot 23}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{12 \pm i \cdot (2 \sqrt{23})}{2 \cdot 1}$

Move $2$ to the left of $i$ .

$x = \frac{12 \pm 2 i \sqrt{23}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{12 \pm 2 i \sqrt{23}}{2}$

Simplify $\frac{12 \pm 2 i \sqrt{23}}{2}$ .

$x = 6 \pm i \sqrt{23}$

Step 4

The final answer is the combination of both solutions.

$x = 6 + i \sqrt{23}, 6 - i \sqrt{23}$