11. \( -3 x^{2}+2 x=15 \)

Solve the equation \( -3x^{2}+2x=15 \). Solve the equation(The complex numbers system) by following steps: - step0: Solve using the quadratic formula in the complex numbers system: \(-3x^{2}+2x=15\) - step1: Move the expression to the left side: \(-3x^{2}+2x-15=0\) - step2: Multiply both sides: \(3x^{2}-2x+15=0\) - step3: Solve using the quadratic formula: \(x=\frac{2\pm \sqrt{\left(-2\right)^{2}-4\times 3\times 15}}{2\times 3}\) - step4: Simplify the expression: \(x=\frac{2\pm \sqrt{\left(-2\right)^{2}-4\times 3\times 15}}{6}\) - step5: Simplify the expression: \(x=\frac{2\pm \sqrt{-176}}{6}\) - step6: Simplify the expression: \(x=\frac{2\pm 4\sqrt{11}\times i}{6}\) - step7: Separate into possible cases: \(\begin{align}&x=\frac{2+4\sqrt{11}\times i}{6}\\&x=\frac{2-4\sqrt{11}\times i}{6}\end{align}\) - step8: Simplify the expression: \(\begin{align}&x=\frac{1}{3}+\frac{2\sqrt{11}}{3}i\\&x=\frac{2-4\sqrt{11}\times i}{6}\end{align}\) - step9: Simplify the expression: \(\begin{align}&x=\frac{1}{3}+\frac{2\sqrt{11}}{3}i\\&x=\frac{1}{3}-\frac{2\sqrt{11}}{3}i\end{align}\) - step10: Rewrite: \(x_{1}=\frac{1}{3}-\frac{2\sqrt{11}}{3}i,x_{2}=\frac{1}{3}+\frac{2\sqrt{11}}{3}i\) - step11: Remove the complex number(s): \(\textrm{No real solution}\) The equation \(-3x^{2}+2x=15\) does not have any real solutions.