Find All Complex Solutions x^2+4x-3.5=0

$x^{2} + 4 x - 3.5 = 0$

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 $a = 1$ , $b = 4$ , and $c = - 3.5$ into the quadratic formula and solve for $x$ .

$\frac{- 4 \pm \sqrt{4^{2} - 4 \cdot (1 \cdot - 3.5)}}{2 \cdot 1}$

Step 3

Simplify.

Simplify the numerator.

Raise $4$ to the power of $2$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot - 3.5}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 3.5$ .

Multiply $- 4$ by $1$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot - 3.5}}{2 \cdot 1}$

Multiply $- 4$ by $- 3.5$ .

$x = \frac{- 4 \pm \sqrt{16 + 14}}{2 \cdot 1}$

Add $16$ and $14$ .

$x = \frac{- 4 \pm \sqrt{30}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 4 \pm \sqrt{30}}{2}$

Step 4

The final answer is the combination of both solutions.

$x = - \frac{4 - \sqrt{30}}{2}, - \frac{4 + \sqrt{30}}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{4 - \sqrt{30}}{2}, - \frac{4 + \sqrt{30}}{2}$

Decimal Form:

$x = 0.73861278 \dots, - 4.73861278 \dots$