Solve Using the Quadratic Formula 3x^2-10x-1=0

$3 x^{2} - 10 x - 1 = 0$

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 $a = 3$ , $b = - 10$ , and $c = - 1$ into the quadratic formula and solve for $x$ .

$\frac{10 \pm \sqrt{{(- 10)}^{2} - 4 \cdot (3 \cdot - 1)}}{2 \cdot 3}$

Step 3

Simplify.

Simplify the numerator.

Raise $- 10$ to the power of $2$ .

$x = \frac{10 \pm \sqrt{100 - 4 \cdot 3 \cdot - 1}}{2 \cdot 3}$

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

Multiply $- 4$ by $3$ .

$x = \frac{10 \pm \sqrt{100 - 12 \cdot - 1}}{2 \cdot 3}$

Multiply $- 12$ by $- 1$ .

$x = \frac{10 \pm \sqrt{100 + 12}}{2 \cdot 3}$

Add $100$ and $12$ .

$x = \frac{10 \pm \sqrt{112}}{2 \cdot 3}$

Rewrite $112$ as $4^{2} \cdot 7$ .

Factor $16$ out of $112$ .

$x = \frac{10 \pm \sqrt{16 (7)}}{2 \cdot 3}$

Rewrite $16$ as $4^{2}$ .

$x = \frac{10 \pm \sqrt{4^{2} \cdot 7}}{2 \cdot 3}$

Pull terms out from under the radical.

$x = \frac{10 \pm 4 \sqrt{7}}{2 \cdot 3}$

Multiply $2$ by $3$ .

$x = \frac{10 \pm 4 \sqrt{7}}{6}$

Simplify $\frac{10 \pm 4 \sqrt{7}}{6}$ .

$x = \frac{5 \pm 2 \sqrt{7}}{3}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{5 \pm 2 \sqrt{7}}{3}$

Decimal Form:

$x = 3.43050087 \dots, - 0.09716754 \dots$