Question

complete the square to indentify the two values that solve the following quadratice equations: x^2+8x=9

Answer 1

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

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

2. **Complete the square:**
- Take the coefficient of \(x\) (which is 8), divide it by 2, and square it:
\[
\left(\frac{8}{2}\right)^2 = 4^2 = 16
\]

- Add and subtract this value inside the equation:
\[
x^2 + 8x + 16 - 16 - 9 = 0
\]

- Rearranging gives:
\[
(x + 4)^2 - 25 = 0
\]

3. **Isolate the squared term:**
\[
(x + 4)^2 = 25
\]

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

5. **Solve for \(x\):**
- For the positive case:
\[
x + 4 = 5 \implies x = 5 - 4 = 1
\]
- For the negative case:
\[
x + 4 = -5 \implies x = -5 - 4 = -9
\]

Thus, the two values that solve the equation \(x^2 + 8x = 9\) are:
\[
\boxed{1} \quad \text{and} \quad \boxed{-9}
\]