Let's solve the equation [tex]\(\sqrt{x+9} - 5 = x + 4\)[/tex] to find its non-extraneous solutions. We'll go through the solution step-by-step.
1. Rewrite the equation:
[tex]\[
\sqrt{x + 9} - 5 = x + 4
\][/tex]
2. Isolate the square root:
[tex]\[
\sqrt{x + 9} = x + 9
\][/tex]
3. Square both sides to remove the square root:
[tex]\[
(\sqrt{x + 9})^2 = (x + 9)^2
\][/tex]
[tex]\[
x + 9 = (x + 9)^2
\][/tex]
4. Expand the right-hand side:
[tex]\[
x + 9 = x^2 + 18x + 81
\][/tex]
5. Bring all terms to one side of the equation to set it to zero:
[tex]\[
0 = x^2 + 18x + 81 - x - 9
\][/tex]
[tex]\[
0 = x^2 + 17x + 72
\][/tex]
6. Solve the quadratic equation using factorization:
[tex]\[
x^2 + 17x + 72 = 0
\][/tex]
Find factors of 72 that sum to 17:
[tex]\[
(x + 8)(x + 9) = 0
\][/tex]
Setting each factor equal to zero gives the potential solutions:
[tex]\[
x + 8 = 0 \quad \text{or} \quad x + 9 = 0
\][/tex]
[tex]\[
x = -8 \quad \text{or} \quad x = -9
\][/tex]
7. Verify each potential solution to check for extraneous solutions:
- For [tex]\(x = -8\)[/tex]:
[tex]\[
\sqrt{-8 + 9} - 5 \stackrel{?}{=} -8 + 4
\][/tex]
[tex]\[
\sqrt{1} - 5 = -4 \quad \text{True}
\][/tex]
- For [tex]\(x = -9\)[/tex]:
[tex]\[
\sqrt{-9 + 9} - 5 \stackrel{?}{=} -9 + 4
\][/tex]
[tex]\[
\sqrt{0} - 5 = -5 \quad \text{True}
\][/tex]
Thus, the non-extraneous solutions are [tex]\(x = -9\)[/tex] and [tex]\(x = -8\)[/tex].
Therefore, the correct answer is:
[tex]\[
(x = -9 \text{ and } x = -8)
\][/tex]
