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

Question

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