Find All Complex Solutions -3x^2+5x+3=0

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

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

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

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

Simplify.

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Simplify the numerator.

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

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

Multiply $- 3$ by $3$ .

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot - 9}}{2 \cdot - 3}$

Multiply $- 4$ by $- 9$ .

$x = \frac{- 5 \pm \sqrt{25 + 36}}{2 \cdot - 3}$

Add $25$ and $36$ .

$x = \frac{- 5 \pm \sqrt{61}}{2 \cdot - 3}$

Multiply $2$ by $- 3$ .

$x = \frac{- 5 \pm \sqrt{61}}{- 6}$

Simplify $\frac{- 5 \pm \sqrt{61}}{- 6}$ .

$x = \frac{5 \pm \sqrt{61}}{6}$

The final answer is the combination of both solutions.

$x = \frac{5 + \sqrt{61}}{6}, \frac{5 - \sqrt{61}}{6}$

The result can be shown in multiple forms.

Exact Form:

$x = \frac{5 + \sqrt{61}}{6}, \frac{5 - \sqrt{61}}{6}$

Decimal Form:

$x = 2.13504161 \dots, - 0.46837494 \dots$

Find All Complex Solutions -3x^2+5x+3=0

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