1. (a) Solve quadratic equations by completing the square. \( \begin{array}{ll}\text { (i) } 10=3 x-x^{2} & \text { (ii) } 4 x^{2}-x=8\end{array} \)

Solve the equation \( 10=3x-x^{2} \). Solve the equation(The complex numbers system) by following steps: - step0: Solve using the quadratic formula in the complex numbers system: \(10=3x-x^{2}\) - step1: Swap the sides: \(3x-x^{2}=10\) - step2: Move the expression to the left side: \(3x-x^{2}-10=0\) - step3: Rewrite in standard form: \(-x^{2}+3x-10=0\) - step4: Multiply both sides: \(x^{2}-3x+10=0\) - step5: Solve using the quadratic formula: \(x=\frac{3\pm \sqrt{\left(-3\right)^{2}-4\times 10}}{2}\) - step6: Simplify the expression: \(x=\frac{3\pm \sqrt{-31}}{2}\) - step7: Simplify the expression: \(x=\frac{3\pm \sqrt{31}\times i}{2}\) - step8: Separate into possible cases: \(\begin{align}&x=\frac{3+\sqrt{31}\times i}{2}\\&x=\frac{3-\sqrt{31}\times i}{2}\end{align}\) - step9: Simplify the expression: \(\begin{align}&x=\frac{3}{2}+\frac{\sqrt{31}}{2}i\\&x=\frac{3-\sqrt{31}\times i}{2}\end{align}\) - step10: Simplify the expression: \(\begin{align}&x=\frac{3}{2}+\frac{\sqrt{31}}{2}i\\&x=\frac{3}{2}-\frac{\sqrt{31}}{2}i\end{align}\) - step11: Rewrite: \(x_{1}=\frac{3}{2}-\frac{\sqrt{31}}{2}i,x_{2}=\frac{3}{2}+\frac{\sqrt{31}}{2}i\) - step12: Remove the complex number(s): \(\textrm{No real solution}\) Solve the equation \( 4x^{2}-x=8 \). Solve the quadratic equation by following steps: - step0: Solve using the quadratic formula: \(4x^{2}-x=8\) - step1: Move the expression to the left side: \(4x^{2}-x-8=0\) - step2: Solve using the quadratic formula: \(x=\frac{1\pm \sqrt{\left(-1\right)^{2}-4\times 4\left(-8\right)}}{2\times 4}\) - step3: Simplify the expression: \(x=\frac{1\pm \sqrt{\left(-1\right)^{2}-4\times 4\left(-8\right)}}{8}\) - step4: Simplify the expression: \(x=\frac{1\pm \sqrt{129}}{8}\) - step5: Separate into possible cases: \(\begin{align}&x=\frac{1+\sqrt{129}}{8}\\&x=\frac{1-\sqrt{129}}{8}\end{align}\) - step6: Rewrite: \(x_{1}=\frac{1-\sqrt{129}}{8},x_{2}=\frac{1+\sqrt{129}}{8}\) The solutions to the quadratic equations by completing the square are: (i) \(10=3x-x^{2}\) has no real solution. (ii) \(4x^{2}-x=8\) has two solutions: \(x_{1}=\frac{1-\sqrt{129}}{8}\) and \(x_{2}=\frac{1+\sqrt{129}}{8}\).