Find All Complex Solutions 2x^2-21x+27=0

$2 x^{2} - 21 x + 27 = 0$

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 $a = 2$ , $b = - 21$ , and $c = 27$ into the quadratic formula and solve for $x$ .

$\frac{21 \pm \sqrt{{(- 21)}^{2} - 4 \cdot (2 \cdot 27)}}{2 \cdot 2}$

Step 3

Simplify.

Simplify the numerator.

Raise $- 21$ to the power of $2$ .

$x = \frac{21 \pm \sqrt{441 - 4 \cdot 2 \cdot 27}}{2 \cdot 2}$

Multiply $- 4 \cdot 2 \cdot 27$ .

Multiply $- 4$ by $2$ .

$x = \frac{21 \pm \sqrt{441 - 8 \cdot 27}}{2 \cdot 2}$

Multiply $- 8$ by $27$ .

$x = \frac{21 \pm \sqrt{441 - 216}}{2 \cdot 2}$

Subtract $216$ from $441$ .

$x = \frac{21 \pm \sqrt{225}}{2 \cdot 2}$

Rewrite $225$ as $15^{2}$ .

$x = \frac{21 \pm \sqrt{15^{2}}}{2 \cdot 2}$

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{21 \pm 15}{2 \cdot 2}$

Multiply $2$ by $2$ .

$x = \frac{21 \pm 15}{4}$

Step 4

The final answer is the combination of both solutions.

$x = 9, \frac{3}{2}$