Solve Using the Quadratic Formula -4x^2+11x+2=0

$- 4 x^{2} + 11 x + 2 = 0$

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 $a = - 4$ , $b = 11$ , and $c = 2$ into the quadratic formula and solve for $x$ .

$\frac{- 11 \pm \sqrt{11^{2} - 4 \cdot (- 4 \cdot 2)}}{2 \cdot - 4}$

Step 3

Simplify.

Simplify the numerator.

Raise $11$ to the power of $2$ .

$x = \frac{- 11 \pm \sqrt{121 - 4 \cdot - 4 \cdot 2}}{2 \cdot - 4}$

Multiply $- 4 \cdot - 4 \cdot 2$ .

Multiply $- 4$ by $- 4$ .

$x = \frac{- 11 \pm \sqrt{121 + 16 \cdot 2}}{2 \cdot - 4}$

Multiply $16$ by $2$ .

$x = \frac{- 11 \pm \sqrt{121 + 32}}{2 \cdot - 4}$

Add $121$ and $32$ .

$x = \frac{- 11 \pm \sqrt{153}}{2 \cdot - 4}$

Rewrite $153$ as $3^{2} \cdot 17$ .

Factor $9$ out of $153$ .

$x = \frac{- 11 \pm \sqrt{9 (17)}}{2 \cdot - 4}$

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

$x = \frac{- 11 \pm \sqrt{3^{2} \cdot 17}}{2 \cdot - 4}$

Pull terms out from under the radical.

$x = \frac{- 11 \pm 3 \sqrt{17}}{2 \cdot - 4}$

Multiply $2$ by $- 4$ .

$x = \frac{- 11 \pm 3 \sqrt{17}}{- 8}$

Simplify $\frac{- 11 \pm 3 \sqrt{17}}{- 8}$ .

$x = \frac{11 \pm 3 \sqrt{17}}{8}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{11 \pm 3 \sqrt{17}}{8}$

Decimal Form:

$x = 2.92116460 \dots, - 0.17116460 \dots$