Question

Solve the equation.


`\sqrt{y^(2) -20y+100 }= y-10`

Answer 1

`\\ \tt\hookrightarrow √(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.