Let's solve the equation [tex]\(\sqrt{x + 10} - 1 = x\)[/tex] step-by-step.
1. Isolate the square root term:
Start by adding 1 to both sides of the equation:
[tex]\[
\sqrt{x + 10} = x + 1
\][/tex]
2. Square both sides:
To eliminate the square root, square both sides of the equation:
[tex]\[
(\sqrt{x + 10})^2 = (x + 1)^2
\][/tex]
This simplifies to:
[tex]\[
x + 10 = (x + 1)^2
\][/tex]
3. Expand the right side:
Expand [tex]\((x + 1)^2\)[/tex]:
[tex]\[
x + 10 = x^2 + 2x + 1
\][/tex]
4. Rearrange the equation:
Finally, move all terms to one side to form a standard quadratic equation:
[tex]\[
x + 10 = x^2 + 2x + 1
\][/tex]
Subtract [tex]\(x + 10\)[/tex] from both sides:
[tex]\[
0 = x^2 + x - 9
\][/tex]
Therefore, the equation [tex]\(\sqrt{x + 10} - 1 = x\)[/tex] simplifies to [tex]\(x + 10 = x^2 + 2x + 1\)[/tex].
Thus, the second equation, [tex]\(x+10 = x^2 + 2x + 1\)[/tex], is the one that matches the transformed form of [tex]\(\sqrt{x + 10} - 1 = x\)[/tex].
