Solve the equation for all values of x by completing the square. Express your answer in simplest form.

Question

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: \[ \boxed{6 \text{ and } -4} \]