35
Solve for $x$ :
\[
\log (x)+\log (x+2)=2
\]

Step 1 ：\(\log (x(x+2))=2\)

Step 2 ：\(\log (x^2+2x)=2\)

Step 3 ：\(10^2 = x^2 + 2x\)

Step 4 ：\(100 = x^2 + 2x\)

Step 5 ：\(x^2 + 2x - 100 = 0\)

Step 6 ：\(x = \frac{-2 \pm \sqrt{(2)^2 - 4(1)(-100)}}{2(1)}\)

Step 7 ：\(x = \frac{-2 \pm \sqrt{4 + 400}}{2}\)

Step 8 ：\(x = \frac{-2 \pm \sqrt{404}}{2}\)

Step 9 ：\(x = -1 \pm \sqrt{101}\)

Step 10 ：Check the solutions in the original equation

Step 11 ：\(x = -1 + \sqrt{101}\) is a valid solution

Step 12 ：\(x = -1 - \sqrt{101}\) is not a valid solution

Step 13 ：\(\boxed{x = -1 + \sqrt{101}}\) is the only solution