Find All Complex Solutions 7x^2+3x-3=0

$7 x^{2} + 3 x - 3 = 0$

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 $a = 7$ , $b = 3$ , and $c = - 3$ into the quadratic formula and solve for $x$ .

$\frac{- 3 \pm \sqrt{3^{2} - 4 \cdot (7 \cdot - 3)}}{2 \cdot 7}$

Step 3

Simplify.

Simplify the numerator.

Raise $3$ to the power of $2$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 7 \cdot - 3}}{2 \cdot 7}$

Multiply $- 4 \cdot 7 \cdot - 3$ .

Multiply $- 4$ by $7$ .

$x = \frac{- 3 \pm \sqrt{9 - 28 \cdot - 3}}{2 \cdot 7}$

Multiply $- 28$ by $- 3$ .

$x = \frac{- 3 \pm \sqrt{9 + 84}}{2 \cdot 7}$

Add $9$ and $84$ .

$x = \frac{- 3 \pm \sqrt{93}}{2 \cdot 7}$

Multiply $2$ by $7$ .

$x = \frac{- 3 \pm \sqrt{93}}{14}$

Step 4

The final answer is the combination of both solutions.

$x = - \frac{3 - \sqrt{93}}{14}, - \frac{3 + \sqrt{93}}{14}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{3 - \sqrt{93}}{14}, - \frac{3 + \sqrt{93}}{14}$

Decimal Form:

$x = 0.47454648 \dots, - 0.90311791 \dots$