Question

Solve the equation for all values of x by completing the square. Express your answer in simplest form.
x, squared, minus, 2, x, equals, 24
x
2
−2x=
24

Answer 1

To solve the equation \( x^2 - 2x = 24 \) by completing the square, follow these steps:

1. Move the constant term to the right side of the equation:
\[
x^2 - 2x - 24 = 0
\]

2. Take the \( x^2 - 2x \) part and complete the square:
- To complete the square, take half of the coefficient of \( x \) (which is \(-2\)), square it, and add to both sides. Half of \(-2\) is \(-1\), and squaring it gives \(1\).
- Add \(1\) to both sides:
\[
x^2 - 2x + 1 = 24 + 1
\]
This simplifies to:
\[
(x - 1)^2 = 25
\]

3. Solve for \( x \) by taking the square root of both sides:
\[
x - 1 = \pm 5
\]

4. Now solve for \( x \):
- For the positive case:
\[
x - 1 = 5 \implies x = 6
\]
- For the negative case:
\[
x - 1 = -5 \implies x = -4
\]

Thus, the solutions to the equation \( x^2 - 2x = 24 \) are:
\[
\boxed{6 \text{ and } -4}
\]