Question

How do you solve `sqrt (100 - d^2) = 10 - d`?

Answer 1

We have `10-d>=0` or `10>=d`

Hence

`sqrt(100-d^2)=10-d`

`sqrt((10+d)(10-d))=sqrt((10-d)^2)`

For `d!=10` we have that

`sqrt((10+d)/(10-d))=1`

Take squares in both sides

`(10+d)/(10-d)=1`

`10+d=10-d`

`d=0`

Hence the solutions are `d=0` and `d=10`

Answer 2

`d = 0` and `10`

Explanation:

`sqrt(100-d^2) = 10 - d`

First, square both sides:
`(sqrt(100-d^2))^2 = (10 - d)^2`

`100 - d^2 = 100 - 20d + d^2`

Subtract `color(blue)100` from both sides of the equation:
`100 - d^2 quadcolor(blue)(-quad100) = 100 - 20d + d^2 quadcolor(blue)(-quad100)`

`-d^2 = -20d + d^2`

Add `color(blue)(d^2)` to both sides of the equation:
`-d^2 quadcolor(blue)(+quadd^2) = -20d + d^2 quadcolor(blue)(+quadd^2)`

`0 = 2d^2 -20d`

Factor out a `color(blue)(2d)`:
`0 = 2d(d-10)`

`2d = 0` and `d - 10 = 0`

`d = 0` and `d = 10`

`-------------------`

Now plug in both solutions to make sure they are really solutions:

First plug in `0`:
`sqrt(100-d^2) = 10 - d`

`sqrt(100-0) = 10 - 0`

`sqrt(100) = 10`

`10 = 10`

This is true. Therefore, `0` is a solution.

Now plug in `10`:
`sqrt(100-d^2) = 10 - d`

`sqrt(100-10^2) = 10 - 10`

`sqrt(100-100)=0`

`sqrt0=0`

`0=0`

This is also a solution.

Hope this helps!