Notice that
y² - 20y + 100 = y² - 2•10y + 10² = (y - 10)²
Then
√(y² - 20y + 100) = y - 10
is equivalent to
√((y - 10)²) = y - 10
|y - 10| = y - 10
If y ≥ 10, then |y - 10| = y - 10, and
y - 10 = y - 10 ⇒ 0 = 0
so there are infinitely many solutions, y ≥ 10.
Otherwise, if y < 10, then |y - 10| = -(y - 10), and
-(y - 10) = y - 10 ⇒ 10 = -10
which is false, so there are no solutions in this case.
[tex]\\ \tt\hookrightarrow \sqrt{y^2-20y+100}=y-10[/tex]
[tex]\\ \tt\hookrightarrow y^2-20y+100=(y-10)^2[/tex]
[tex]\\ \tt\hookrightarrow y^2-20y+100=y^2-20y+100[/tex]
Both sides are equal
Hence
