Find All Complex Solutions 6x^2=90

$6 x^{2} = 90$

Step 1

Subtract $90$ from both sides of the equation.

$6 x^{2} - 90 = 0$

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 $a = 6$ , $b = 0$ , and $c = - 90$ into the quadratic formula and solve for $x$ .

$\frac{0 \pm \sqrt{0^{2} - 4 \cdot (6 \cdot - 90)}}{2 \cdot 6}$

Step 4

Simplify.

Simplify the numerator.

Raising $0$ to any positive power yields $0$ .

$x = \frac{0 \pm \sqrt{0 - 4 \cdot 6 \cdot - 90}}{2 \cdot 6}$

Multiply $- 4 \cdot 6 \cdot - 90$ .

Multiply $- 4$ by $6$ .

$x = \frac{0 \pm \sqrt{0 - 24 \cdot - 90}}{2 \cdot 6}$

Multiply $- 24$ by $- 90$ .

$x = \frac{0 \pm \sqrt{0 + 2160}}{2 \cdot 6}$

Add $0$ and $2160$ .

$x = \frac{0 \pm \sqrt{2160}}{2 \cdot 6}$

Rewrite $2160$ as $12^{2} \cdot 15$ .

Factor $144$ out of $2160$ .

$x = \frac{0 \pm \sqrt{144 (15)}}{2 \cdot 6}$

Rewrite $144$ as $12^{2}$ .

$x = \frac{0 \pm \sqrt{12^{2} \cdot 15}}{2 \cdot 6}$

Pull terms out from under the radical.

$x = \frac{0 \pm 12 \sqrt{15}}{2 \cdot 6}$

Multiply $2$ by $6$ .

$x = \frac{0 \pm 12 \sqrt{15}}{12}$

Simplify $\frac{0 \pm 12 \sqrt{15}}{12}$ .

$x = \pm \sqrt{15}$

Step 5

The final answer is the combination of both solutions.

$x = \sqrt{15}, - \sqrt{15}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \sqrt{15}, - \sqrt{15}$

Decimal Form:

$x = 3.87298334 \dots, - 3.87298334 \dots$