Find All Complex Solutions 3x^2-24x+9=0

$3 x^{2} - 24 x + 9 = 0$

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 $a = 3$ , $b = - 24$ , and $c = 9$ into the quadratic formula and solve for $x$ .

$\frac{24 \pm \sqrt{{(- 24)}^{2} - 4 \cdot (3 \cdot 9)}}{2 \cdot 3}$

Step 3

Simplify.

Simplify the numerator.

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

$x = \frac{24 \pm \sqrt{576 - 4 \cdot 3 \cdot 9}}{2 \cdot 3}$

Multiply $- 4 \cdot 3 \cdot 9$ .

Multiply $- 4$ by $3$ .

$x = \frac{24 \pm \sqrt{576 - 12 \cdot 9}}{2 \cdot 3}$

Multiply $- 12$ by $9$ .

$x = \frac{24 \pm \sqrt{576 - 108}}{2 \cdot 3}$

Subtract $108$ from $576$ .

$x = \frac{24 \pm \sqrt{468}}{2 \cdot 3}$

Rewrite $468$ as $6^{2} \cdot 13$ .

Factor $36$ out of $468$ .

$x = \frac{24 \pm \sqrt{36 (13)}}{2 \cdot 3}$

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

$x = \frac{24 \pm \sqrt{6^{2} \cdot 13}}{2 \cdot 3}$

Pull terms out from under the radical.

$x = \frac{24 \pm 6 \sqrt{13}}{2 \cdot 3}$

Multiply $2$ by $3$ .

$x = \frac{24 \pm 6 \sqrt{13}}{6}$

Simplify $\frac{24 \pm 6 \sqrt{13}}{6}$ .

$x = 4 \pm \sqrt{13}$

Step 4

The final answer is the combination of both solutions.

$x = 4 + \sqrt{13}, 4 - \sqrt{13}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = 4 + \sqrt{13}, 4 - \sqrt{13}$

Decimal Form:

$x = 7.60555127 \dots, 0.39444872 \dots$