Solve the equation using the quadratic formula. \[ x^{2}-12 x+61=0 \] The solution set is \( \} \). (Simplity your answer. Type an exact answer, using radicals and \( i \) as needed. Use integers or fractions for any numbers in the expression. Use a comma to se answers as needed.)

Solve the equation(The complex numbers system) by following steps: - step0: Solve using the quadratic formula in the complex numbers system: \(x^{2}-12x+61=0\) - step1: Solve using the quadratic formula: \(x=\frac{12\pm \sqrt{\left(-12\right)^{2}-4\times 61}}{2}\) - step2: Simplify the expression: \(x=\frac{12\pm \sqrt{-100}}{2}\) - step3: Simplify the expression: \(x=\frac{12\pm 10i}{2}\) - step4: Separate into possible cases: \(\begin{align}&x=\frac{12+10i}{2}\\&x=\frac{12-10i}{2}\end{align}\) - step5: Simplify the expression: \(\begin{align}&x=6+5i\\&x=\frac{12-10i}{2}\end{align}\) - step6: Simplify the expression: \(\begin{align}&x=6+5i\\&x=6-5i\end{align}\) - step7: Rewrite: \(x_{1}=6-5i,x_{2}=6+5i\) To solve the equation \[ x^{2}-12x+61=0 \] using the quadratic formula, we identify the coefficients: - \( a = 1 \) - \( b = -12 \) - \( c = 61 \) The quadratic formula is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Now, we calculate the discriminant \( b^2 - 4ac \): \[ b^2 - 4ac = (-12)^2 - 4 \cdot 1 \cdot 61 = 144 - 244 = -100 \] Since the discriminant is negative, we will have complex solutions. Now we can substitute the values into the quadratic formula: \[ x = \frac{-(-12) \pm \sqrt{-100}}{2 \cdot 1} = \frac{12 \pm 10i}{2} \] Simplifying this gives: \[ x = 6 \pm 5i \] Thus, the solution set is \[ \{ 6 - 5i, 6 + 5i \} \]