Solve Using the Quadratic Formula 3x^2-x-1=0

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

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

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

Step 3

Simplify.

Simplify the numerator.

Raise $- 1$ to the power of $2$ .

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

Multiply $- 4 \cdot 3 \cdot - 1$ .

Multiply $- 4$ by $3$ .

$x = \frac{1 \pm \sqrt{1 - 12 \cdot - 1}}{2 \cdot 3}$

Multiply $- 12$ by $- 1$ .

$x = \frac{1 \pm \sqrt{1 + 12}}{2 \cdot 3}$

Add $1$ and $12$ .

$x = \frac{1 \pm \sqrt{13}}{2 \cdot 3}$

Multiply $2$ by $3$ .

$x = \frac{1 \pm \sqrt{13}}{6}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1 \pm \sqrt{13}}{6}$

Decimal Form:

$x = 0.76759187 \dots, - 0.43425854 \dots$