Question

Complete the square to identify the two values that solve the following quadratic equation: x² + 8x = 9.
X = __

Answer 1

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

1. Move the constant to the right side of the equation:
\[
x^2 + 8x - 9 = 0
\]

2. Rearrange it to the form suitable for completing the square:
\[
x^2 + 8x = 9
\]

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

4. Add and subtract this square (16) to the left side of the equation:
\[
x^2 + 8x + 16 - 16 = 9
\]
This can be rewritten as:
\[
(x + 4)^2 - 16 = 9
\]

5. Now, isolate the squared term:
\[
(x + 4)^2 = 9 + 16
\]
\[
(x + 4)^2 = 25
\]

6. Take the square root of both sides:
\[
x + 4 = 5 \quad \text{or} \quad x + 4 = -5
\]

7. Solve for \( x \) in each case:
\[
x + 4 = 5 \implies x = 5 - 4 = 1
\]
\[
x + 4 = -5 \implies x = -5 - 4 = -9
\]

Therefore, the two solutions to the equation \( x^2 + 8x = 9 \) are:
\[
x = 1 \quad \text{and} \quad x = -9
\]

Final answer:
\[
X = 1, -9
\]