Solve Using the Quadratic Formula 9x^2-6x+1=7

$9 x^{2} - 6 x + 1 = 7$

Step 1

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

Subtract $7$ from both sides of the equation.

$9 x^{2} - 6 x + 1 - 7 = 0$

Subtract $7$ from $1$ .

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

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

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

Step 4

Simplify.

Simplify the numerator.

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

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

Multiply $- 4 \cdot 9 \cdot - 6$ .

Multiply $- 4$ by $9$ .

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

Multiply $- 36$ by $- 6$ .

$x = \frac{6 \pm \sqrt{36 + 216}}{2 \cdot 9}$

Add $36$ and $216$ .

$x = \frac{6 \pm \sqrt{252}}{2 \cdot 9}$

Rewrite $252$ as $6^{2} \cdot 7$ .

Factor $36$ out of $252$ .

$x = \frac{6 \pm \sqrt{36 (7)}}{2 \cdot 9}$

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

$x = \frac{6 \pm \sqrt{6^{2} \cdot 7}}{2 \cdot 9}$

Pull terms out from under the radical.

$x = \frac{6 \pm 6 \sqrt{7}}{2 \cdot 9}$

Multiply $2$ by $9$ .

$x = \frac{6 \pm 6 \sqrt{7}}{18}$

Simplify $\frac{6 \pm 6 \sqrt{7}}{18}$ .

$x = \frac{1 \pm \sqrt{7}}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1 \pm \sqrt{7}}{3}$

Decimal Form:

$x = 1.21525043 \dots, - 0.54858377 \dots$