To solve the given equation \( x^2 + 16x = 22 \) and find an equivalent one, we will complete the square. Here is a step-by-step solution:
1. Start with the given equation:
[tex]\[
x^2 + 16x = 22
\][/tex]
2. Complete the square on the left-hand side:
- Take the coefficient of \( x \), which is 16.
- Divide it by 2:
[tex]\[
\frac{16}{2} = 8
\][/tex]
- Square the result:
[tex]\[
8^2 = 64
\][/tex]
3. Add and subtract 64 to the equation to maintain equality:
[tex]\[
x^2 + 16x + 64 - 64 = 22
\][/tex]
4. Rewrite the left side as a perfect square:
[tex]\[
(x + 8)^2 - 64 = 22
\][/tex]
5. Move the constant term (-64) to the right-hand side of the equation:
[tex]\[
(x + 8)^2 = 22 + 64
\][/tex]
6. Combine the terms on the right-hand side:
[tex]\[
(x + 8)^2 = 86
\][/tex]
Therefore, the equation that is equivalent to the given equation \( x^2 + 16x = 22 \) is:
A. \((x + 8)^2 = 86\)
So, the correct answer is:
[tex]\[
\boxed{(x+8)^2=86}
\][/tex]
