2.1. Solve for \( x \) 2.1.1. \( \quad 2 x^{2}-12 x=-32 \)

Solve the equation \( 2x^{2}-12x=-32 \). Solve the equation(The complex numbers system) by following steps: - step0: Solve using the quadratic formula in the complex numbers system: \(2x^{2}-12x=-32\) - step1: Move the expression to the left side: \(2x^{2}-12x+32=0\) - step2: Solve using the quadratic formula: \(x=\frac{12\pm \sqrt{\left(-12\right)^{2}-4\times 2\times 32}}{2\times 2}\) - step3: Simplify the expression: \(x=\frac{12\pm \sqrt{\left(-12\right)^{2}-4\times 2\times 32}}{4}\) - step4: Simplify the expression: \(x=\frac{12\pm \sqrt{-112}}{4}\) - step5: Simplify the expression: \(x=\frac{12\pm 4\sqrt{7}\times i}{4}\) - step6: Separate into possible cases: \(\begin{align}&x=\frac{12+4\sqrt{7}\times i}{4}\\&x=\frac{12-4\sqrt{7}\times i}{4}\end{align}\) - step7: Simplify the expression: \(\begin{align}&x=3+\sqrt{7}\times i\\&x=\frac{12-4\sqrt{7}\times i}{4}\end{align}\) - step8: Simplify the expression: \(\begin{align}&x=3+\sqrt{7}\times i\\&x=3-\sqrt{7}\times i\end{align}\) - step9: Rewrite: \(x_{1}=3-\sqrt{7}\times i,x_{2}=3+\sqrt{7}\times i\) The solutions to the equation \(2x^{2}-12x=-32\) are \(x_{1}=3-\sqrt{7}i\) and \(x_{2}=3+\sqrt{7}i\).