1. \( 6 x^{2}-x+2=0 \) 2. \( x^{2}-3 x+7=0 \) 3. \( m^{2}+8 m+16=0 \) 4. \( 3 x^{2}+8 x-8=0 \) 5. \( 3 x^{2}-4 x+2=0 \)

Solve the equation \( 6x^{2}-x+2=0 \). Solve the equation(The complex numbers system) by following steps: - step0: Solve using the quadratic formula in the complex numbers system: \(6x^{2}-x+2=0\) - step1: Solve using the quadratic formula: \(x=\frac{1\pm \sqrt{\left(-1\right)^{2}-4\times 6\times 2}}{2\times 6}\) - step2: Simplify the expression: \(x=\frac{1\pm \sqrt{\left(-1\right)^{2}-4\times 6\times 2}}{12}\) - step3: Simplify the expression: \(x=\frac{1\pm \sqrt{-47}}{12}\) - step4: Simplify the expression: \(x=\frac{1\pm \sqrt{47}\times i}{12}\) - step5: Separate into possible cases: \(\begin{align}&x=\frac{1+\sqrt{47}\times i}{12}\\&x=\frac{1-\sqrt{47}\times i}{12}\end{align}\) - step6: Simplify the expression: \(\begin{align}&x=\frac{1}{12}+\frac{\sqrt{47}}{12}i\\&x=\frac{1-\sqrt{47}\times i}{12}\end{align}\) - step7: Simplify the expression: \(\begin{align}&x=\frac{1}{12}+\frac{\sqrt{47}}{12}i\\&x=\frac{1}{12}-\frac{\sqrt{47}}{12}i\end{align}\) - step8: Rewrite: \(x_{1}=\frac{1}{12}-\frac{\sqrt{47}}{12}i,x_{2}=\frac{1}{12}+\frac{\sqrt{47}}{12}i\) - step9: Remove the complex number(s): \(\textrm{No real solution}\) Solve the equation \( x^{2}-3x+7=0 \). Solve the equation(The complex numbers system) by following steps: - step0: Solve using the quadratic formula in the complex numbers system: \(x^{2}-3x+7=0\) - step1: Solve using the quadratic formula: \(x=\frac{3\pm \sqrt{\left(-3\right)^{2}-4\times 7}}{2}\) - step2: Simplify the expression: \(x=\frac{3\pm \sqrt{-19}}{2}\) - step3: Simplify the expression: \(x=\frac{3\pm \sqrt{19}\times i}{2}\) - step4: Separate into possible cases: \(\begin{align}&x=\frac{3+\sqrt{19}\times i}{2}\\&x=\frac{3-\sqrt{19}\times i}{2}\end{align}\) - step5: Simplify the expression: \(\begin{align}&x=\frac{3}{2}+\frac{\sqrt{19}}{2}i\\&x=\frac{3-\sqrt{19}\times i}{2}\end{align}\) - step6: Simplify the expression: \(\begin{align}&x=\frac{3}{2}+\frac{\sqrt{19}}{2}i\\&x=\frac{3}{2}-\frac{\sqrt{19}}{2}i\end{align}\) - step7: Rewrite: \(x_{1}=\frac{3}{2}-\frac{\sqrt{19}}{2}i,x_{2}=\frac{3}{2}+\frac{\sqrt{19}}{2}i\) - step8: Remove the complex number(s): \(\textrm{No real solution}\) Solve the equation \( m^{2}+8m+16=0 \). Solve the quadratic equation by following steps: - step0: Solve by factoring: \(m^{2}+8m+16=0\) - step1: Factor the expression: \(\left(m+4\right)^{2}=0\) - step2: Simplify the expression: \(m+4=0\) - step3: Move the constant to the right side: \(m=0-4\) - step4: Remove 0: \(m=-4\) Solve the equation \( 3x^{2}+8x-8=0 \). Solve the quadratic equation by following steps: - step0: Solve using the quadratic formula: \(3x^{2}+8x-8=0\) - step1: Solve using the quadratic formula: \(x=\frac{-8\pm \sqrt{8^{2}-4\times 3\left(-8\right)}}{2\times 3}\) - step2: Simplify the expression: \(x=\frac{-8\pm \sqrt{8^{2}-4\times 3\left(-8\right)}}{6}\) - step3: Simplify the expression: \(x=\frac{-8\pm \sqrt{160}}{6}\) - step4: Simplify the expression: \(x=\frac{-8\pm 4\sqrt{10}}{6}\) - step5: Separate into possible cases: \(\begin{align}&x=\frac{-8+4\sqrt{10}}{6}\\&x=\frac{-8-4\sqrt{10}}{6}\end{align}\) - step6: Simplify the expression: \(\begin{align}&x=\frac{-4+2\sqrt{10}}{3}\\&x=\frac{-8-4\sqrt{10}}{6}\end{align}\) - step7: Simplify the expression: \(\begin{align}&x=\frac{-4+2\sqrt{10}}{3}\\&x=-\frac{4+2\sqrt{10}}{3}\end{align}\) - step8: Rewrite: \(x_{1}=-\frac{4+2\sqrt{10}}{3},x_{2}=\frac{-4+2\sqrt{10}}{3}\) Solve the equation \( 3x^{2}-4x+2=0 \). Solve the equation(The complex numbers system) by following steps: - step0: Solve using the quadratic formula in the complex numbers system: \(3x^{2}-4x+2=0\) - step1: Solve using the quadratic formula: \(x=\frac{4\pm \sqrt{\left(-4\right)^{2}-4\times 3\times 2}}{2\times 3}\) - step2: Simplify the expression: \(x=\frac{4\pm \sqrt{\left(-4\right)^{2}-4\times 3\times 2}}{6}\) - step3: Simplify the expression: \(x=\frac{4\pm \sqrt{-8}}{6}\) - step4: Simplify the expression: \(x=\frac{4\pm 2\sqrt{2}\times i}{6}\) - step5: Separate into possible cases: \(\begin{align}&x=\frac{4+2\sqrt{2}\times i}{6}\\&x=\frac{4-2\sqrt{2}\times i}{6}\end{align}\) - step6: Simplify the expression: \(\begin{align}&x=\frac{2}{3}+\frac{\sqrt{2}}{3}i\\&x=\frac{4-2\sqrt{2}\times i}{6}\end{align}\) - step7: Simplify the expression: \(\begin{align}&x=\frac{2}{3}+\frac{\sqrt{2}}{3}i\\&x=\frac{2}{3}-\frac{\sqrt{2}}{3}i\end{align}\) - step8: Rewrite: \(x_{1}=\frac{2}{3}-\frac{\sqrt{2}}{3}i,x_{2}=\frac{2}{3}+\frac{\sqrt{2}}{3}i\) - step9: Remove the complex number(s): \(\textrm{No real solution}\) The solutions to the given equations are as follows: 1. \(6x^{2}-x+2=0\) has no real solutions. 2. \(x^{2}-3x+7=0\) has no real solutions. 3. \(m^{2}+8m+16=0\) has a solution \(m=-4\). 4. \(3x^{2}+8x-8=0\) has solutions \(x_{1}=-\frac{4+2\sqrt{10}}{3}\) and \(x_{2}=\frac{-4+2\sqrt{10}}{3}\). 5. \(3x^{2}-4x+2=0\) has no real solutions.