Find All Complex Solutions 2x^2-2=-0.5x+4

$2 x^{2} - 2 = - 0.5 x + 4$

Step 1

Move all terms to the left side of the equation and simplify.

Move all the expressions to the left side of the equation.

Add $0.5 x$ to both sides of the equation.

$2 x^{2} - 2 + 0.5 x = 4$

Subtract $4$ from both sides of the equation.

$2 x^{2} - 2 + 0.5 x - 4 = 0$

Subtract $4$ from $- 2$ .

$2 x^{2} + 0.5 x - 6 = 0$

Step 2

Use the quadratic formula to find the solutions.

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

Step 3

Substitute the values $a = 2$ , $b = 0.5$ , and $c = - 6$ into the quadratic formula and solve for $x$ .

$\frac{- 0.5 \pm \sqrt{{0.5}^{2} - 4 \cdot (2 \cdot - 6)}}{2 \cdot 2}$

Step 4

Simplify.

Simplify the numerator.

Raise $0.5$ to the power of $2$ .

$x = \frac{- 0.5 \pm \sqrt{0.25 - 4 \cdot 2 \cdot - 6}}{2 \cdot 2}$

Multiply $- 4 \cdot 2 \cdot - 6$ .

Multiply $- 4$ by $2$ .

$x = \frac{- 0.5 \pm \sqrt{0.25 - 8 \cdot - 6}}{2 \cdot 2}$

Multiply $- 8$ by $- 6$ .

$x = \frac{- 0.5 \pm \sqrt{0.25 + 48}}{2 \cdot 2}$

Add $0.25$ and $48$ .

$x = \frac{- 0.5 \pm \sqrt{48.25}}{2 \cdot 2}$

Multiply $2$ by $2$ .

$x = \frac{- 0.5 \pm \sqrt{48.25}}{4}$

Step 5

The final answer is the combination of both solutions.

$x = 1.61155549, - 1.86155549$