Solve for x 2x^2-5x-1=0

$2 x^{2} - 5 x - 1 = 0$

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 $a = 2$ , $b = - 5$ , and $c = - 1$ into the quadratic formula and solve for $x$ .

$\frac{5 \pm \sqrt{{(- 5)}^{2} - 4 \cdot (2 \cdot - 1)}}{2 \cdot 2}$

Step 3

Simplify.

Simplify the numerator.

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

$x = \frac{5 \pm \sqrt{25 - 4 \cdot 2 \cdot - 1}}{2 \cdot 2}$

Multiply $- 4 \cdot 2 \cdot - 1$ .

Multiply $- 4$ by $2$ .

$x = \frac{5 \pm \sqrt{25 - 8 \cdot - 1}}{2 \cdot 2}$

Multiply $- 8$ by $- 1$ .

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

Add $25$ and $8$ .

$x = \frac{5 \pm \sqrt{33}}{2 \cdot 2}$

Multiply $2$ by $2$ .

$x = \frac{5 \pm \sqrt{33}}{4}$

Step 4

The final answer is the combination of both solutions.

$x = \frac{5 + \sqrt{33}}{4}, \frac{5 - \sqrt{33}}{4}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{5 + \sqrt{33}}{4}, \frac{5 - \sqrt{33}}{4}$

Decimal Form:

$x = 2.68614066 \dots, - 0.18614066 \dots$