Solve Using the Quadratic Formula -9x^2+6x-1=0

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

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 $a = - 9$ , $b = 6$ , and $c = - 1$ into the quadratic formula and solve for $x$ .

$\frac{- 6 \pm \sqrt{6^{2} - 4 \cdot (- 9 \cdot - 1)}}{2 \cdot - 9}$

Step 3

Simplify.

Simplify the numerator.

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

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot - 9 \cdot - 1}}{2 \cdot - 9}$

Multiply $- 4 \cdot - 9 \cdot - 1$ .

Multiply $- 4$ by $- 9$ .

$x = \frac{- 6 \pm \sqrt{36 + 36 \cdot - 1}}{2 \cdot - 9}$

Multiply $36$ by $- 1$ .

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

Subtract $36$ from $36$ .

$x = \frac{- 6 \pm \sqrt{0}}{2 \cdot - 9}$

Rewrite $0$ as $0^{2}$ .

$x = \frac{- 6 \pm \sqrt{0^{2}}}{2 \cdot - 9}$

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

$x = \frac{- 6 \pm 0}{2 \cdot - 9}$

$- 6$ plus or minus $0$ is $- 6$ .

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

Multiply $2$ by $- 9$ .

$x = \frac{- 6}{- 18}$

Cancel the common factor of $- 6$ and $- 18$ .

Factor $- 6$ out of $- 6$ .

$x = \frac{- 6 \cdot 1}{- 18}$

Cancel the common factors.

Factor $- 6$ out of $- 18$ .

$x = \frac{- 6 \cdot 1}{- 6 \cdot 3}$

Cancel the common factor.

$x = \frac{- 6 \cdot 1}{- 6 \cdot 3}$

Rewrite the expression.

$x = \frac{1}{3}$

Step 4

The final answer is the combination of both solutions.

$x = \frac{1}{3}$ Double roots