\( 1 \in \) Solve the equation using the quadratic formula. \[ x^{2}-4 x+40=0 \] 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 separate 0 of 1.5 .73 answers as needed.)

Solve the equation(The complex numbers system) by following steps: - step0: Solve using the quadratic formula in the complex numbers system: \(x^{2}-4x+40=0\) - step1: Solve using the quadratic formula: \(x=\frac{4\pm \sqrt{\left(-4\right)^{2}-4\times 40}}{2}\) - step2: Simplify the expression: \(x=\frac{4\pm \sqrt{-144}}{2}\) - step3: Simplify the expression: \(x=\frac{4\pm 12i}{2}\) - step4: Separate into possible cases: \(\begin{align}&x=\frac{4+12i}{2}\\&x=\frac{4-12i}{2}\end{align}\) - step5: Simplify the expression: \(\begin{align}&x=2+6i\\&x=\frac{4-12i}{2}\end{align}\) - step6: Simplify the expression: \(\begin{align}&x=2+6i\\&x=2-6i\end{align}\) - step7: Rewrite: \(x_{1}=2-6i,x_{2}=2+6i\) To solve the equation \( x^{2}-4x+40=0 \) using the quadratic formula, we can identify the coefficients: - \( a = 1 \) - \( b = -4 \) - \( c = 40 \) The quadratic formula is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Now, let's calculate the discriminant \( b^2 - 4ac \): \[ b^2 - 4ac = (-4)^2 - 4 \cdot 1 \cdot 40 = 16 - 160 = -144 \] Since the discriminant is negative, we will have complex solutions. Now we can substitute the values into the quadratic formula: \[ x = \frac{-(-4) \pm \sqrt{-144}}{2 \cdot 1} = \frac{4 \pm \sqrt{-144}}{2} \] We know that \( \sqrt{-144} = 12i \), so we can continue: \[ x = \frac{4 \pm 12i}{2} = 2 \pm 6i \] Thus, the solutions are: \[ x_{1} = 2 - 6i, \quad x_{2} = 2 + 6i \] The solution set is: \[ \{ 2 - 6i, 2 + 6i \} \]