Solve for x 5x^2-5x-3=0

$5 x^{2} - 5 x - 3 = 0$

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 $a = 5$ , $b = - 5$ , and $c = - 3$ into the quadratic formula and solve for $x$ .

$\frac{5 \pm \sqrt{{(- 5)}^{2} - 4 \cdot (5 \cdot - 3)}}{2 \cdot 5}$

Step 3

Simplify.

Simplify the numerator.

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

$x = \frac{5 \pm \sqrt{25 - 4 \cdot 5 \cdot - 3}}{2 \cdot 5}$

Multiply $- 4 \cdot 5 \cdot - 3$ .

Multiply $- 4$ by $5$ .

$x = \frac{5 \pm \sqrt{25 - 20 \cdot - 3}}{2 \cdot 5}$

Multiply $- 20$ by $- 3$ .

$x = \frac{5 \pm \sqrt{25 + 60}}{2 \cdot 5}$

Add $25$ and $60$ .

$x = \frac{5 \pm \sqrt{85}}{2 \cdot 5}$

Multiply $2$ by $5$ .

$x = \frac{5 \pm \sqrt{85}}{10}$

Step 4

The final answer is the combination of both solutions.

$x = \frac{5 + \sqrt{85}}{10}, \frac{5 - \sqrt{85}}{10}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{5 + \sqrt{85}}{10}, \frac{5 - \sqrt{85}}{10}$

Decimal Form:

$x = 1.42195444 \dots, - 0.42195444 \dots$