Question

Solve the equation." title="\sqrt{y^{2} -20y+100 }= y-10" alt="\sqrt{y^{2} -20y+100 }= y-10" align="absmiddle" class="latex-formula">

Answer 1

$\\ \tt\hookrightarrow \sqrt{y^2-20y+100}=y-10$

$\\ \tt\hookrightarrow y^2-20y+100=(y-10)^2$

$\\ \tt\hookrightarrow y^2-20y+100=y^2-20y+100$

Both sides are equal

Hence

• No solution

Answer 2

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.