Question 2 \( 2.12 x^{2}+2 x+1=0 \) solve for \( x \)

Solve the equation \( 2.12x^2+2x+1=0 \). Solve the equation(The complex numbers system) by following steps: - step0: Solve using the quadratic formula in the complex numbers system: \(2.12x^{2}+2x+1=0\) - step1: Convert the decimal into a fraction: \(\frac{53}{25}x^{2}+2x+1=0\) - step2: Multiply both sides: \(25\left(\frac{53}{25}x^{2}+2x+1\right)=25\times 0\) - step3: Calculate: \(53x^{2}+50x+25=0\) - step4: Solve using the quadratic formula: \(x=\frac{-50\pm \sqrt{50^{2}-4\times 53\times 25}}{2\times 53}\) - step5: Simplify the expression: \(x=\frac{-50\pm \sqrt{50^{2}-4\times 53\times 25}}{106}\) - step6: Simplify the expression: \(x=\frac{-50\pm \sqrt{-2800}}{106}\) - step7: Simplify the expression: \(x=\frac{-50\pm 20\sqrt{7}\times i}{106}\) - step8: Separate into possible cases: \(\begin{align}&x=\frac{-50+20\sqrt{7}\times i}{106}\\&x=\frac{-50-20\sqrt{7}\times i}{106}\end{align}\) - step9: Simplify the expression: \(\begin{align}&x=-\frac{25}{53}+\frac{10\sqrt{7}}{53}i\\&x=\frac{-50-20\sqrt{7}\times i}{106}\end{align}\) - step10: Simplify the expression: \(\begin{align}&x=-\frac{25}{53}+\frac{10\sqrt{7}}{53}i\\&x=-\frac{25}{53}-\frac{10\sqrt{7}}{53}i\end{align}\) - step11: Rewrite: \(x_{1}=-\frac{25}{53}-\frac{10\sqrt{7}}{53}i,x_{2}=-\frac{25}{53}+\frac{10\sqrt{7}}{53}i\) The solutions to the equation \(2.12x^2+2x+1=0\) are: \[x_{1}=-\frac{25}{53}-\frac{10\sqrt{7}}{53}i\] \[x_{2}=-\frac{25}{53}+\frac{10\sqrt{7}}{53}i\]