Complete the square to identify the two values that solve the following quadratic equation: x^2 + 8x = 9.

Question

x = ___

Answer 1

To solve the quadratic equation \( x^2 + 8x = 9 \) by completing the square, follow these steps:

1. Rearrange the equation to set it to zero: \[ x^2 + 8x - 9 = 0 \]

2. Move the constant term to the other side: \[ x^2 + 8x = 9 \]

3. To complete the square, take half of the coefficient of \( x \) (which is 8), square it, and add it to both sides. Half of 8 is 4, and squaring it gives: \[ 4^2 = 16 \]

4. Add 16 to both sides of the equation: \[ x^2 + 8x + 16 = 9 + 16 \] \[ x^2 + 8x + 16 = 25 \]

5. Now, the left side is a perfect square: \[ (x + 4)^2 = 25 \]

6. Take the square root of both sides: \[ x + 4 = \pm 5 \]

7. Solve for \( x \):

   • \( x + 4 = 5 \): \[ x = 5 - 4 = 1 \]
   • \( x + 4 = -5 \): \[ x = -5 - 4 = -9 \]

Thus, the two values that solve the quadratic equation are: \[ x = 1 \quad \text{and} \quad x = -9 \]

So, the answer is: \[ x = 1, -9 \]