Find All Complex Solutions -7x^2-23x+10=0

$- 7 x^{2} - 23 x + 10 = 0$

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 $a = - 7$ , $b = - 23$ , and $c = 10$ into the quadratic formula and solve for $x$ .

$\frac{23 \pm \sqrt{{(- 23)}^{2} - 4 \cdot (- 7 \cdot 10)}}{2 \cdot - 7}$

Step 3

Simplify.

Simplify the numerator.

Raise $- 23$ to the power of $2$ .

$x = \frac{23 \pm \sqrt{529 - 4 \cdot - 7 \cdot 10}}{2 \cdot - 7}$

Multiply $- 4 \cdot - 7 \cdot 10$ .

Multiply $- 4$ by $- 7$ .

$x = \frac{23 \pm \sqrt{529 + 28 \cdot 10}}{2 \cdot - 7}$

Multiply $28$ by $10$ .

$x = \frac{23 \pm \sqrt{529 + 280}}{2 \cdot - 7}$

Add $529$ and $280$ .

$x = \frac{23 \pm \sqrt{809}}{2 \cdot - 7}$

Multiply $2$ by $- 7$ .

$x = \frac{23 \pm \sqrt{809}}{- 14}$

Move the negative in front of the fraction.

$x = - \frac{23 \pm \sqrt{809}}{14}$

Step 4

The final answer is the combination of both solutions.

$x = - \frac{23 + \sqrt{809}}{14}, - \frac{23 - \sqrt{809}}{14}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{23 + \sqrt{809}}{14}, - \frac{23 - \sqrt{809}}{14}$

Decimal Form:

$x = - 3.67449466 \dots, 0.38878037 \dots$