Question

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

Answer 1

To solve the quadratic equation \( x^2 + 8x = 9 \) by completing the square, we first rearrange the equation to set it to zero:

\[
x^2 + 8x - 9 = 0
\]

Next, we need to complete the square for the expression \( x^2 + 8x \). To do this, we take the coefficient of \( x \), which is 8, divide it by 2, and then square it:

\[
\left(\frac{8}{2}\right)^2 = 4^2 = 16
\]

Now we can rewrite \( x^2 + 8x \) as a perfect square. We add and subtract 16 in the equation:

\[
x^2 + 8x + 16 - 16 - 9 = 0
\]

This simplifies to:

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

Now we can isolate the perfect square:

\[
(x + 4)^2 = 25
\]

Next, we take the square root of both sides:

\[
x + 4 = \pm 5
\]

This leads to two equations:

1. \( x + 4 = 5 \)
2. \( x + 4 = -5 \)

Solving these equations gives us:

1. \( x = 5 - 4 \) which simplifies to \( x = 1 \)
2. \( x = -5 - 4 \) which simplifies to \( x = -9 \)

Thus, the two values that solve the quadratic equation \( x^2 + 8x = 9 \) are:

\[
\boxed{1} \quad \text{and} \quad \boxed{-9}
\]