Let's solve the equation step by step.
Given equation:
[tex]\[
\sqrt{(x+9)^2} = \pm \sqrt{25}
\][/tex]
To solve this, we need to consider both possible scenarios separately:
### Case 1: [tex]\( \sqrt{(x+9)^2} = \sqrt{25} \)[/tex]
In this case, we have:
[tex]\[
(x+9)^2 = 25
\][/tex]
Taking the square root on both sides (considering the principal square root):
[tex]\[
x + 9 = 5
\][/tex]
Now, solving for [tex]\( x \)[/tex]:
[tex]\[
x = 5 - 9
\][/tex]
[tex]\[
x = -4
\][/tex]
### Case 2: [tex]\( \sqrt{(x+9)^2} = -\sqrt{25} \)[/tex]
In this case, we have:
[tex]\[
(x+9)^2 = 25
\][/tex]
Taking the square root on both sides (considering the negative square root):
[tex]\[
x + 9 = -5
\][/tex]
Now, solving for [tex]\( x \)[/tex]:
[tex]\[
x = -5 - 9
\][/tex]
[tex]\[
x = -14
\][/tex]
Hence, the solutions to the equation are:
If [tex]\( x+9=5 \)[/tex], then [tex]\( x= -4 \)[/tex].
If [tex]\( x+9=-5 \)[/tex], then [tex]\( x= -14 \)[/tex].
