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, 48
x
2
−2x=
48

Answer 1

To solve the equation \( x^2 - 2x = 48 \) by completing the square, follow these steps:

1. Move the constant to the other side: \[ x^2 - 2x - 48 = 0 \] We can actually keep it as \( x^2 - 2x = 48 \) for completing the square.

2. Complete the square: Take the coefficient of \( x \) (which is -2), halve it, and square it: \[ \left(-\frac{2}{2}\right)^2 = (-1)^2 = 1 \]

3. Add and subtract this square inside the equation: \[ x^2 - 2x + 1 - 1 = 48 \] This can be rewritten as: \[ (x - 1)^2 - 1 = 48 \]

4. Move the constant back to the right side: \[ (x - 1)^2 = 48 + 1 \] \[ (x - 1)^2 = 49 \]

5. Take the square root of both sides: \[ x - 1 = \pm 7 \]

6. Solve for \( x \):

   • If \( x - 1 = 7 \): \[ x = 7 + 1 = 8 \]
   • If \( x - 1 = -7 \): \[ x = -7 + 1 = -6 \]

Thus, the solutions for the equation \( x^2 - 2x = 48 \) are: \[ \boxed{8} \text{ and } \boxed{-6} \]