Solve for x 4x^2+5x-2=0

$4 x^{2} + 5 x - 2 = 0$

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 $a = 4$ , $b = 5$ , and $c = - 2$ into the quadratic formula and solve for $x$ .

$\frac{- 5 \pm \sqrt{5^{2} - 4 \cdot (4 \cdot - 2)}}{2 \cdot 4}$

Step 3

Simplify.

Simplify the numerator.

Raise $5$ to the power of $2$ .

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot 4 \cdot - 2}}{2 \cdot 4}$

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

Multiply $- 4$ by $4$ .

$x = \frac{- 5 \pm \sqrt{25 - 16 \cdot - 2}}{2 \cdot 4}$

Multiply $- 16$ by $- 2$ .

$x = \frac{- 5 \pm \sqrt{25 + 32}}{2 \cdot 4}$

Add $25$ and $32$ .

$x = \frac{- 5 \pm \sqrt{57}}{2 \cdot 4}$

Multiply $2$ by $4$ .

$x = \frac{- 5 \pm \sqrt{57}}{8}$

Step 4

The final answer is the combination of both solutions.

$x = - \frac{5 - \sqrt{57}}{8}, - \frac{5 + \sqrt{57}}{8}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{5 - \sqrt{57}}{8}, - \frac{5 + \sqrt{57}}{8}$

Decimal Form:

$x = 0.31872930 \dots, - 1.56872930 \dots$