Solve for q 9q^2-60q+100=20

$9 q^{2} - 60 q + 100 = 20$

Move all terms to the left side of the equation and simplify.

Subtract $20$ from both sides of the equation.

$9 q^{2} - 60 q + 100 - 20 = 0$

Subtract $20$ from $100$ .

$9 q^{2} - 60 q + 80 = 0$

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Substitute the values $a = 9$ , $b = - 60$ , and $c = 80$ into the quadratic formula and solve for $q$ .

$\frac{60 \pm \sqrt{{(- 60)}^{2} - 4 \cdot (9 \cdot 80)}}{2 \cdot 9}$

Simplify.

Simplify the numerator.

Raise $- 60$ to the power of $2$ .

$q = \frac{60 \pm \sqrt{3600 - 4 \cdot 9 \cdot 80}}{2 \cdot 9}$

Multiply $- 4$ by $9$ .

$q = \frac{60 \pm \sqrt{3600 - 36 \cdot 80}}{2 \cdot 9}$

Multiply $- 36$ by $80$ .

$q = \frac{60 \pm \sqrt{3600 - 2880}}{2 \cdot 9}$

Subtract $2880$ from $3600$ .

$q = \frac{60 \pm \sqrt{720}}{2 \cdot 9}$

Rewrite $720$ as $12^{2} \cdot 5$ .

Factor $144$ out of $720$ .

$q = \frac{60 \pm \sqrt{144 (5)}}{2 \cdot 9}$

Rewrite $144$ as $12^{2}$ .

$q = \frac{60 \pm \sqrt{12^{2} \cdot 5}}{2 \cdot 9}$

Pull terms out from under the radical.

$q = \frac{60 \pm 12 \sqrt{5}}{2 \cdot 9}$

Multiply $2$ by $9$ .

$q = \frac{60 \pm 12 \sqrt{5}}{18}$

Simplify $\frac{60 \pm 12 \sqrt{5}}{18}$ .

$q = \frac{10 \pm 2 \sqrt{5}}{3}$

The final answer is the combination of both solutions.

$q = \frac{10 + 2 \sqrt{5}}{3}, \frac{10 - 2 \sqrt{5}}{3}$

The result can be shown in multiple forms.

Exact Form:

$q = \frac{10 + 2 \sqrt{5}}{3}, \frac{10 - 2 \sqrt{5}}{3}$

Decimal Form:

$q = 4.82404531 \dots, 1.84262134 \dots$