Question

Solve the equation for all values of x by completing the square. Express your answer in simplest form.
x, squared, minus, 8, x, plus, 7, equals, 0
x
2
−8x+7=
0

Answer 1

To solve the equation \( x^2 - 8x + 7 = 0 \) by completing the square, follow these steps:

1. Start with the original equation:
\[
x^2 - 8x + 7 = 0
\]

2. Move the constant term to the other side of the equation:
\[
x^2 - 8x = -7
\]

3. To complete the square, take the coefficient of \( x \) (which is -8), divide it by 2, and square the result:
\[
\left(\frac{-8}{2}\right)^2 = (-4)^2 = 16
\]

4. Add this square to both sides of the equation:
\[
x^2 - 8x + 16 = -7 + 16
\]
This simplifies to:
\[
x^2 - 8x + 16 = 9
\]

5. The left side of the equation can be factored as a square:
\[
(x - 4)^2 = 9
\]

6. Now, take the square root of both sides:
\[
x - 4 = \pm 3
\]

7. Solve for \( x \) by isolating it:
- For the positive case:
\[
x - 4 = 3 \implies x = 3 + 4 = 7
\]
- For the negative case:
\[
x - 4 = -3 \implies x = -3 + 4 = 1
\]

8. Thus, the solutions are:
\[
x = 7 \quad \text{and} \quad x = 1
\]

**Final answer:**
\[
x = 7 \quad \text{or} \quad x = 1
\]