Find All Complex Solutions -5x^2+7x-1=0

$- 5 x^{2} + 7 x - 1 = 0$

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 $a = - 5$ , $b = 7$ , and $c = - 1$ into the quadratic formula and solve for $x$ .

$\frac{- 7 \pm \sqrt{7^{2} - 4 \cdot (- 5 \cdot - 1)}}{2 \cdot - 5}$

Step 3

Simplify.

Simplify the numerator.

Raise $7$ to the power of $2$ .

$x = \frac{- 7 \pm \sqrt{49 - 4 \cdot - 5 \cdot - 1}}{2 \cdot - 5}$

Multiply $- 4 \cdot - 5 \cdot - 1$ .

Multiply $- 4$ by $- 5$ .

$x = \frac{- 7 \pm \sqrt{49 + 20 \cdot - 1}}{2 \cdot - 5}$

Multiply $20$ by $- 1$ .

$x = \frac{- 7 \pm \sqrt{49 - 20}}{2 \cdot - 5}$

Subtract $20$ from $49$ .

$x = \frac{- 7 \pm \sqrt{29}}{2 \cdot - 5}$

Multiply $2$ by $- 5$ .

$x = \frac{- 7 \pm \sqrt{29}}{- 10}$

Simplify $\frac{- 7 \pm \sqrt{29}}{- 10}$ .

$x = \frac{7 \pm \sqrt{29}}{10}$

Step 4

The final answer is the combination of both solutions.

$x = \frac{7 + \sqrt{29}}{10}, \frac{7 - \sqrt{29}}{10}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{7 + \sqrt{29}}{10}, \frac{7 - \sqrt{29}}{10}$

Decimal Form:

$x = 1.23851648 \dots, 0.16148351 \dots$