Certainly! Let's solve the equation [tex]\(\sqrt{x + 9} - x = 9\)[/tex] step-by-step and verify the solutions.
1. Start with the given equation:
[tex]\[
\sqrt{x + 9} - x = 9
\][/tex]
2. Isolate the square root on one side of the equation:
[tex]\[
\sqrt{x + 9} = x + 9
\][/tex]
3. Square both sides of the equation to eliminate the square root:
[tex]\[
(\sqrt{x + 9})^2 = (x + 9)^2
\][/tex]
4. Simplify both sides:
[tex]\[
x + 9 = (x + 9)^2
\][/tex]
5. Expand the right side of the equation:
[tex]\[
x + 9 = x^2 + 18x + 81
\][/tex]
6. Bring all terms to one side to set the equation to zero:
[tex]\[
0 = x^2 + 18x + 81 - x - 9
\][/tex]
7. Simplify the equation:
[tex]\[
0 = x^2 + 17x + 72
\][/tex]
8. Solve the quadratic equation [tex]\(x^2 + 17x + 72 = 0\)[/tex].
The quadratic formula is given by [tex]\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex]. Here, [tex]\(a = 1\)[/tex], [tex]\(b = 17\)[/tex], and [tex]\(c = 72\)[/tex].
[tex]\[
x = \frac{-17 \pm \sqrt{17^2 - 4 \cdot 1 \cdot 72}}{2 \cdot 1}
\][/tex]
[tex]\[
x = \frac{-17 \pm \sqrt{289 - 288}}{2}
\][/tex]
[tex]\[
x = \frac{-17 \pm \sqrt{1}}{2}
\][/tex]
[tex]\[
x = \frac{-17 \pm 1}{2}
\][/tex]
Therefore, the solutions are:
[tex]\[
x = \frac{-17 + 1}{2} = \frac{-16}{2} = -8
\][/tex]
and
[tex]\[
x = \frac{-17 - 1}{2} = \frac{-18}{2} = -9
\][/tex]
9. Verify the solutions:
- For [tex]\(x = -8\)[/tex]:
[tex]\[
\sqrt{-8 + 9} - (-8) = \sqrt{1} + 8 = 1 + 8 = 9
\][/tex]
This is true.
- For [tex]\(x = -9\)[/tex]:
[tex]\[
\sqrt{-9 + 9} - (-9) = \sqrt{0} + 9 = 0 + 9 = 9
\][/tex]
This is also true.
Thus, the solutions to the equation [tex]\(\sqrt{x + 9} - x = 9\)[/tex] are [tex]\(\boxed{-9}\)[/tex] and [tex]\(\boxed{-8}\)[/tex].
