Solve Using the Quadratic Formula 9x^2-30x=-22

$9 x^{2} - 30 x = - 22$

Step 1

Add $22$ to both sides of the equation.

$9 x^{2} - 30 x + 22 = 0$

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 $a = 9$ , $b = - 30$ , and $c = 22$ into the quadratic formula and solve for $x$ .

$\frac{30 \pm \sqrt{{(- 30)}^{2} - 4 \cdot (9 \cdot 22)}}{2 \cdot 9}$

Step 4

Simplify.

Simplify the numerator.

Raise $- 30$ to the power of $2$ .

$x = \frac{30 \pm \sqrt{900 - 4 \cdot 9 \cdot 22}}{2 \cdot 9}$

Multiply $- 4 \cdot 9 \cdot 22$ .

Multiply $- 4$ by $9$ .

$x = \frac{30 \pm \sqrt{900 - 36 \cdot 22}}{2 \cdot 9}$

Multiply $- 36$ by $22$ .

$x = \frac{30 \pm \sqrt{900 - 792}}{2 \cdot 9}$

Subtract $792$ from $900$ .

$x = \frac{30 \pm \sqrt{108}}{2 \cdot 9}$

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

Factor $36$ out of $108$ .

$x = \frac{30 \pm \sqrt{36 (3)}}{2 \cdot 9}$

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

$x = \frac{30 \pm \sqrt{6^{2} \cdot 3}}{2 \cdot 9}$

Pull terms out from under the radical.

$x = \frac{30 \pm 6 \sqrt{3}}{2 \cdot 9}$

Multiply $2$ by $9$ .

$x = \frac{30 \pm 6 \sqrt{3}}{18}$

Simplify $\frac{30 \pm 6 \sqrt{3}}{18}$ .

$x = \frac{5 \pm \sqrt{3}}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{5 \pm \sqrt{3}}{3}$

Decimal Form:

$x = 2.24401693 \dots, 1.08931639 \dots$