Find All Complex Solutions 18x^2-24x+87=0

$18 x^{2} - 24 x + 87 = 0$

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 $a = 18$ , $b = - 24$ , and $c = 87$ into the quadratic formula and solve for $x$ .

$\frac{24 \pm \sqrt{{(- 24)}^{2} - 4 \cdot (18 \cdot 87)}}{2 \cdot 18}$

Step 3

Simplify.

Simplify the numerator.

Raise $- 24$ to the power of $2$ .

$x = \frac{24 \pm \sqrt{576 - 4 \cdot 18 \cdot 87}}{2 \cdot 18}$

Multiply $- 4 \cdot 18 \cdot 87$ .

Multiply $- 4$ by $18$ .

$x = \frac{24 \pm \sqrt{576 - 72 \cdot 87}}{2 \cdot 18}$

Multiply $- 72$ by $87$ .

$x = \frac{24 \pm \sqrt{576 - 6264}}{2 \cdot 18}$

Subtract $6264$ from $576$ .

$x = \frac{24 \pm \sqrt{- 5688}}{2 \cdot 18}$

Rewrite $- 5688$ as $- 1 (5688)$ .

$x = \frac{24 \pm \sqrt{- 1 \cdot 5688}}{2 \cdot 18}$

Rewrite $\sqrt{- 1 (5688)}$ as $\sqrt{- 1} \cdot \sqrt{5688}$ .

$x = \frac{24 \pm \sqrt{- 1} \cdot \sqrt{5688}}{2 \cdot 18}$

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

$x = \frac{24 \pm i \cdot \sqrt{5688}}{2 \cdot 18}$

Rewrite $5688$ as $6^{2} \cdot 158$ .

Factor $36$ out of $5688$ .

$x = \frac{24 \pm i \cdot \sqrt{36 (158)}}{2 \cdot 18}$

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

$x = \frac{24 \pm i \cdot \sqrt{6^{2} \cdot 158}}{2 \cdot 18}$

Pull terms out from under the radical.

$x = \frac{24 \pm i \cdot (6 \sqrt{158})}{2 \cdot 18}$

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

$x = \frac{24 \pm 6 i \sqrt{158}}{2 \cdot 18}$

Multiply $2$ by $18$ .

$x = \frac{24 \pm 6 i \sqrt{158}}{36}$

Simplify $\frac{24 \pm 6 i \sqrt{158}}{36}$ .

$x = \frac{4 \pm i \sqrt{158}}{6}$