Question

1. solve the equation using the quadratic formula.

$x^2 - 6x - 27 = 0$
the solution set is
(simplify your answer. type an exact answer, using rad
use a comma to separate answers as needed.)

Explanation:

Step1: Identify coefficients

For quadratic equation \(ax^2 + bx + c = 0\), here \(a = 1\), \(b = -6\), \(c = -27\).

Step2: Recall quadratic formula

Quadratic formula: \(x=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a}\)

Step3: Calculate discriminant

Discriminant \(D = b^2 - 4ac = (-6)^2 - 4\times1\times(-27)= 36 + 108 = 144\)

Step4: Substitute into formula

\(x=\frac{-(-6)\pm\sqrt{144}}{2\times1}=\frac{6\pm12}{2}\)

Step5: Find two solutions

First solution: \(x_1=\frac{6 + 12}{2}=\frac{18}{2}=9\)
Second solution: \(x_2=\frac{6 - 12}{2}=\frac{-6}{2}=-3\)

Answer:

\(9, -3\)