Find All Complex Solutions x^2-6x=15

$x^{2} - 6 x = 15$

Step 1

Subtract $15$ from both sides of the equation.

$x^{2} - 6 x - 15 = 0$

Step 2

Use the quadratic formula to find the solutions.

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

Step 3

Substitute the values $a = 1$ , $b = - 6$ , and $c = - 15$ into the quadratic formula and solve for $x$ .

$\frac{6 \pm \sqrt{{(- 6)}^{2} - 4 \cdot (1 \cdot - 15)}}{2 \cdot 1}$

Step 4

Simplify.

Simplify the numerator.

Raise $- 6$ to the power of $2$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 1 \cdot - 15}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 15$ .

Multiply $- 4$ by $1$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot - 15}}{2 \cdot 1}$

Multiply $- 4$ by $- 15$ .

$x = \frac{6 \pm \sqrt{36 + 60}}{2 \cdot 1}$

Add $36$ and $60$ .

$x = \frac{6 \pm \sqrt{96}}{2 \cdot 1}$

Rewrite $96$ as $4^{2} \cdot 6$ .

Factor $16$ out of $96$ .

$x = \frac{6 \pm \sqrt{16 (6)}}{2 \cdot 1}$

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

$x = \frac{6 \pm \sqrt{4^{2} \cdot 6}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{6 \pm 4 \sqrt{6}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{6 \pm 4 \sqrt{6}}{2}$

Simplify $\frac{6 \pm 4 \sqrt{6}}{2}$ .

$x = 3 \pm 2 \sqrt{6}$

Step 5

The final answer is the combination of both solutions.

$x = 3 + 2 \sqrt{6}, 3 - 2 \sqrt{6}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = 3 + 2 \sqrt{6}, 3 - 2 \sqrt{6}$

Decimal Form:

$x = 7.89897948 \dots, - 1.89897948 \dots$