To determine which perfect square equation represents [tex]\( x^2 + 10x + 23 = 0 \)[/tex] by completing the square, follow these steps:
1. Start with the given equation:
[tex]\[
x^2 + 10x + 23 = 0
\][/tex]
2. Move the constant term to the other side of the equation:
[tex]\[
x^2 + 10x = -23
\][/tex]
3. Complete the square on the left-hand side:
- Take the coefficient of [tex]\( x \)[/tex], which is 10, and halve it to get 5.
- Square this result to get [tex]\( 5^2 = 25 \)[/tex].
Add and subtract 25 on the left-hand side to keep the equation balanced:
[tex]\[
x^2 + 10x + 25 - 25 = -23
\][/tex]
4. Rewrite the equation:
[tex]\[
(x + 5)^2 - 25 = -23
\][/tex]
5. Simplify the equation by moving [tex]\( -25 \)[/tex] to the right-hand side:
[tex]\[
(x + 5)^2 = 25 - 23
\][/tex]
6. Final simplification yields:
[tex]\[
(x + 5)^2 = 2
\][/tex]
The perfect square equation representing [tex]\( x^2 + 10x + 23 = 0 \)[/tex] by completing the square is:
[tex]\[
(x+5)^2 = 2
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{(x+5)^2=2}
\][/tex]
