Solve by Factoring x^2+3x+4=0

$x^{2} + 3 x + 4 = 0$

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 $a = 1$ , $b = 3$ , and $c = 4$ into the quadratic formula and solve for $x$ .

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

Step 3

Simplify.

Simplify the numerator.

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

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 1 \cdot 4}}{2 \cdot 1}$

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

Multiply $- 4$ by $1$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 4}}{2 \cdot 1}$

Multiply $- 4$ by $4$ .

$x = \frac{- 3 \pm \sqrt{9 - 16}}{2 \cdot 1}$

Subtract $16$ from $9$ .

$x = \frac{- 3 \pm \sqrt{- 7}}{2 \cdot 1}$

Rewrite $- 7$ as $- 1 (7)$ .

$x = \frac{- 3 \pm \sqrt{- 1 \cdot 7}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (7)}$ as $\sqrt{- 1} \cdot \sqrt{7}$ .

$x = \frac{- 3 \pm \sqrt{- 1} \cdot \sqrt{7}}{2 \cdot 1}$

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

$x = \frac{- 3 \pm i \sqrt{7}}{2 \cdot 1}$

Multiply $2$ by $1$ .

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

Step 4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Simplify the numerator.

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

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 1 \cdot 4}}{2 \cdot 1}$

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

Multiply $- 4$ by $1$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 4}}{2 \cdot 1}$

Multiply $- 4$ by $4$ .

$x = \frac{- 3 \pm \sqrt{9 - 16}}{2 \cdot 1}$

Subtract $16$ from $9$ .

$x = \frac{- 3 \pm \sqrt{- 7}}{2 \cdot 1}$

Rewrite $- 7$ as $- 1 (7)$ .

$x = \frac{- 3 \pm \sqrt{- 1 \cdot 7}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (7)}$ as $\sqrt{- 1} \cdot \sqrt{7}$ .

$x = \frac{- 3 \pm \sqrt{- 1} \cdot \sqrt{7}}{2 \cdot 1}$

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

$x = \frac{- 3 \pm i \sqrt{7}}{2 \cdot 1}$

Multiply $2$ by $1$ .

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

Change the $\pm$ to $+$ .

$x = \frac{- 3 + i \sqrt{7}}{2}$

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{- 1 \cdot 3 + i \sqrt{7}}{2}$

Factor $- 1$ out of $i \sqrt{7}$ .

$x = \frac{- 1 \cdot 3 - (- i \sqrt{7})}{2}$

Factor $- 1$ out of $- 1 (3) - (- i \sqrt{7})$ .

$x = \frac{- 1 (3 - i \sqrt{7})}{2}$

Move the negative in front of the fraction.

$x = - \frac{3 - i \sqrt{7}}{2}$

Step 5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Simplify the numerator.

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

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 1 \cdot 4}}{2 \cdot 1}$

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

Multiply $- 4$ by $1$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 4}}{2 \cdot 1}$

Multiply $- 4$ by $4$ .

$x = \frac{- 3 \pm \sqrt{9 - 16}}{2 \cdot 1}$

Subtract $16$ from $9$ .

$x = \frac{- 3 \pm \sqrt{- 7}}{2 \cdot 1}$

Rewrite $- 7$ as $- 1 (7)$ .

$x = \frac{- 3 \pm \sqrt{- 1 \cdot 7}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (7)}$ as $\sqrt{- 1} \cdot \sqrt{7}$ .

$x = \frac{- 3 \pm \sqrt{- 1} \cdot \sqrt{7}}{2 \cdot 1}$

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

$x = \frac{- 3 \pm i \sqrt{7}}{2 \cdot 1}$

Multiply $2$ by $1$ .

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

Change the $\pm$ to $-$ .

$x = \frac{- 3 - i \sqrt{7}}{2}$

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{- 1 \cdot 3 - i \sqrt{7}}{2}$

Factor $- 1$ out of $- i \sqrt{7}$ .

$x = \frac{- 1 \cdot 3 - (i \sqrt{7})}{2}$

Factor $- 1$ out of $- 1 (3) - (i \sqrt{7})$ .

$x = \frac{- 1 (3 + i \sqrt{7})}{2}$

Move the negative in front of the fraction.

$x = - \frac{3 + i \sqrt{7}}{2}$

Step 6

The final answer is the combination of both solutions.

$x = - \frac{3 - i \sqrt{7}}{2}, - \frac{3 + i \sqrt{7}}{2}$

Solve by Factoring x^2+3x+4=0

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