39 8. \( -2 x^{2}+2 x=5 \)

Solve the equation \( -2x^{2}+2x=5 \). Solve the equation(The complex numbers system) by following steps: - step0: Solve using the quadratic formula in the complex numbers system: \(-2x^{2}+2x=5\) - step1: Move the expression to the left side: \(-2x^{2}+2x-5=0\) - step2: Multiply both sides: \(2x^{2}-2x+5=0\) - step3: Solve using the quadratic formula: \(x=\frac{2\pm \sqrt{\left(-2\right)^{2}-4\times 2\times 5}}{2\times 2}\) - step4: Simplify the expression: \(x=\frac{2\pm \sqrt{\left(-2\right)^{2}-4\times 2\times 5}}{4}\) - step5: Simplify the expression: \(x=\frac{2\pm \sqrt{-36}}{4}\) - step6: Simplify the expression: \(x=\frac{2\pm 6i}{4}\) - step7: Separate into possible cases: \(\begin{align}&x=\frac{2+6i}{4}\\&x=\frac{2-6i}{4}\end{align}\) - step8: Simplify the expression: \(\begin{align}&x=\frac{1}{2}+\frac{3}{2}i\\&x=\frac{2-6i}{4}\end{align}\) - step9: Simplify the expression: \(\begin{align}&x=\frac{1}{2}+\frac{3}{2}i\\&x=\frac{1}{2}-\frac{3}{2}i\end{align}\) - step10: Rewrite: \(x_{1}=\frac{1}{2}-\frac{3}{2}i,x_{2}=\frac{1}{2}+\frac{3}{2}i\) - step11: Remove the complex number(s): \(\textrm{No real solution}\) The equation \(-2x^{2}+2x=5\) does not have any real solutions.