Solve Using the Quadratic Formula 2x^2-6x=9

$2 x^{2} - 6 x = 9$

Step 1

Subtract $9$ from both sides of the equation.

$2 x^{2} - 6 x - 9 = 0$

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 $a = 2$ , $b = - 6$ , and $c = - 9$ into the quadratic formula and solve for $x$ .

$\frac{6 \pm \sqrt{{(- 6)}^{2} - 4 \cdot (2 \cdot - 9)}}{2 \cdot 2}$

Step 4

Simplify.

Simplify the numerator.

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

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 2 \cdot - 9}}{2 \cdot 2}$

Multiply $- 4 \cdot 2 \cdot - 9$ .

Multiply $- 4$ by $2$ .

$x = \frac{6 \pm \sqrt{36 - 8 \cdot - 9}}{2 \cdot 2}$

Multiply $- 8$ by $- 9$ .

$x = \frac{6 \pm \sqrt{36 + 72}}{2 \cdot 2}$

Add $36$ and $72$ .

$x = \frac{6 \pm \sqrt{108}}{2 \cdot 2}$

Rewrite $108$ as $6^{2} \cdot 3$ .

Factor $36$ out of $108$ .

$x = \frac{6 \pm \sqrt{36 (3)}}{2 \cdot 2}$

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

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

Pull terms out from under the radical.

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

Multiply $2$ by $2$ .

$x = \frac{6 \pm 6 \sqrt{3}}{4}$

Simplify $\frac{6 \pm 6 \sqrt{3}}{4}$ .

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 4.09807621 \dots, - 1.09807621 \dots$