Solve for x 9x^2+3x-1=0

$9 x^{2} + 3 x - 1 = 0$

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 $a = 9$ , $b = 3$ , and $c = - 1$ into the quadratic formula and solve for $x$ .

$\frac{- 3 \pm \sqrt{3^{2} - 4 \cdot (9 \cdot - 1)}}{2 \cdot 9}$

Step 3

Simplify.

Simplify the numerator.

Raise $3$ to the power of $2$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 9 \cdot - 1}}{2 \cdot 9}$

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

Multiply $- 4$ by $9$ .

$x = \frac{- 3 \pm \sqrt{9 - 36 \cdot - 1}}{2 \cdot 9}$

Multiply $- 36$ by $- 1$ .

$x = \frac{- 3 \pm \sqrt{9 + 36}}{2 \cdot 9}$

Add $9$ and $36$ .

$x = \frac{- 3 \pm \sqrt{45}}{2 \cdot 9}$

Rewrite $45$ as $3^{2} \cdot 5$ .

Factor $9$ out of $45$ .

$x = \frac{- 3 \pm \sqrt{9 (5)}}{2 \cdot 9}$

Rewrite $9$ as $3^{2}$ .

$x = \frac{- 3 \pm \sqrt{3^{2} \cdot 5}}{2 \cdot 9}$

Pull terms out from under the radical.

$x = \frac{- 3 \pm 3 \sqrt{5}}{2 \cdot 9}$

Multiply $2$ by $9$ .

$x = \frac{- 3 \pm 3 \sqrt{5}}{18}$

Simplify $\frac{- 3 \pm 3 \sqrt{5}}{18}$ .

$x = \frac{- 1 \pm \sqrt{5}}{6}$

Step 4

The final answer is the combination of both solutions.

$x = - \frac{1 - \sqrt{5}}{6}, - \frac{1 + \sqrt{5}}{6}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{1 - \sqrt{5}}{6}, - \frac{1 + \sqrt{5}}{6}$

Decimal Form:

$x = 0.20601132 \dots, - 0.53934466 \dots$