Solve Using the Quadratic Formula -3x^2+4x+1=0

$- 3 x^{2} + 4 x + 1 = 0$

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 $a = - 3$ , $b = 4$ , and $c = 1$ into the quadratic formula and solve for $x$ .

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

Step 3

Simplify.

Simplify the numerator.

Raise $4$ to the power of $2$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot - 3 \cdot 1}}{2 \cdot - 3}$

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

Multiply $- 4$ by $- 3$ .

$x = \frac{- 4 \pm \sqrt{16 + 12 \cdot 1}}{2 \cdot - 3}$

Multiply $12$ by $1$ .

$x = \frac{- 4 \pm \sqrt{16 + 12}}{2 \cdot - 3}$

Add $16$ and $12$ .

$x = \frac{- 4 \pm \sqrt{28}}{2 \cdot - 3}$

Rewrite $28$ as $2^{2} \cdot 7$ .

Factor $4$ out of $28$ .

$x = \frac{- 4 \pm \sqrt{4 (7)}}{2 \cdot - 3}$

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

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

Pull terms out from under the radical.

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

Multiply $2$ by $- 3$ .

$x = \frac{- 4 \pm 2 \sqrt{7}}{- 6}$

Simplify $\frac{- 4 \pm 2 \sqrt{7}}{- 6}$ .

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

Step 4

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 1.54858377 \dots, - 0.21525043 \dots$