Find All Complex Solutions -2x^2-5x+4=0

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

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

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

Step 3

Simplify.

Simplify the numerator.

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

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

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

Multiply $- 4$ by $- 2$ .

$x = \frac{5 \pm \sqrt{25 + 8 \cdot 4}}{2 \cdot - 2}$

Multiply $8$ by $4$ .

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

Add $25$ and $32$ .

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

Multiply $2$ by $- 2$ .

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

Move the negative in front of the fraction.

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

Step 4

The final answer is the combination of both solutions.

$x = - \frac{5 + \sqrt{57}}{4}, - \frac{5 - \sqrt{57}}{4}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{5 + \sqrt{57}}{4}, - \frac{5 - \sqrt{57}}{4}$

Decimal Form:

$x = - 3.13745860 \dots, 0.63745860 \dots$