Solve Using the Quadratic Formula x^2+x+3=0

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

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 $a = 1$ , $b = 1$ , and $c = 3$ into the quadratic formula and solve for $x$ .

$\frac{- 1 \pm \sqrt{1^{2} - 4 \cdot (1 \cdot 3)}}{2 \cdot 1}$

Step 3

Simplify.

Simplify the numerator.

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1 \cdot 3}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 3$ .

Multiply $- 4$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 3}}{2 \cdot 1}$

Multiply $- 4$ by $3$ .

$x = \frac{- 1 \pm \sqrt{1 - 12}}{2 \cdot 1}$

Subtract $12$ from $1$ .

$x = \frac{- 1 \pm \sqrt{- 11}}{2 \cdot 1}$

Rewrite $- 11$ as $- 1 (11)$ .

$x = \frac{- 1 \pm \sqrt{- 1 \cdot 11}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (11)}$ as $\sqrt{- 1} \cdot \sqrt{11}$ .

$x = \frac{- 1 \pm \sqrt{- 1} \cdot \sqrt{11}}{2 \cdot 1}$

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 1 \pm i \sqrt{11}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 1 \pm i \sqrt{11}}{2}$