To solve the quadratic equation [tex]\(3x^2 - 6x - 24 = 0\)[/tex] by completing the square, follow these steps:
1. Move the constant term to the right side of the equation:
[tex]\[
3x^2 - 6x = 24
\][/tex]
2. Divide every term by the coefficient of [tex]\(x^2\)[/tex] (which is 3):
[tex]\[
\frac{3x^2}{3} - \frac{6x}{3} = \frac{24}{3}
\][/tex]
[tex]\[
x^2 - 2x = 8
\][/tex]
3. Add and subtract the square of half the coefficient of [tex]\(x\)[/tex] inside the equation:
The coefficient of [tex]\(x\)[/tex] is -2, so half of -2 is -1, and the square of -1 is 1. Add and subtract this square inside the equation:
[tex]\[
x^2 - 2x + 1 - 1 = 8
\][/tex]
[tex]\[
(x - 1)^2 - 1 = 8
\][/tex]
4. Simplify and solve for [tex]\((x - 1)^2\)[/tex]:
[tex]\[
(x - 1)^2 - 1 + 1 = 8 + 1
\][/tex]
[tex]\[
(x - 1)^2 = 9
\][/tex]
5. Take the square root of both sides to solve for [tex]\(x - 1\)[/tex]:
[tex]\[
x - 1 = \pm \sqrt{9}
\][/tex]
[tex]\[
x - 1 = 3 \quad \text{or} \quad x - 1 = -3
\][/tex]
6. Solve for [tex]\(x\)[/tex]:
[tex]\[
x = 3 + 1 = 4 \quad \text{or} \quad x = -3 + 1 = -2
\][/tex]
Thus, the solutions to the equation [tex]\(3x^2 - 6x - 24 = 0\)[/tex] are:
[tex]\[
x = 4 \quad \text{and} \quad x = -2
\][/tex]
These match the solution pair given in the problem, which is [tex]\((4, -2)\)[/tex].
