Question

`sqrt (t) = sqrt (t - 12) + 2?`solve the radical equations, of possible.

Answer 1

THIS ANSWER IS INCORRECT. SEE THE CORRECT SOLUTION ABOVE.

Explanation:

Start by squaring both sides to get rid of one of the radicals, then simplify and combine like terms.
`sqrtt^color(green)2=(sqrt(t-12)+2)^color(green)2`
`t=t-12+4sqrt(t-12)+4`
`t=t-8+4sqrt(t-12)`

Then operate on both sides of the equation to isolate the other radical.
`tcolor(green)(-t)=color(red)cancelcolor(black)t-8+4sqrt(t-12)color(red)cancelcolor(green)(-t)`
`0color(green)(+8)=color(red)cancelcolor(black)("-"8)+4sqrt(t-12)color(red)cancelcolor(green)(+8)`
`color(green)(color(black)8/4)=color(green)((color(red)cancelcolor(black)4color(black)sqrt(t-12))/color(red)cancelcolor(green)4`
`8=sqrt(t-12)`

And square both sides again to get rid of the other radical.
`8^color(green)2=sqrt(t-12)^color(green)2`
`64=t-12`

Finally, add `12` to both sides to isolate `t`.
`64color(green)(+12)=tcolor(red)cancelcolor(black)(-12)color(red)cancelcolor(green)(+12)`
`76=t`
`t=76`

When working with radicals, always check your solutions to make sure they aren't extraneous (make sure they don't cause there to be a square root of a negative number). In this case both `76` and `76-12` are positive, so `76` is a valid solution for `t`.

Answer 2

`x in {16}`

Explanation:

Rearrange the equation:

`sqrt(t) - 2 = sqrt(t - 12)`

Square both sides:

`(sqrt(t) - 2)^2 = (sqrt(t - 12))^2`

`t - 4sqrt(t) + 4 = t - 12`

Simplify:

`16 = 4sqrt(t)`

`4 = sqrt(t)`

Square both sides once more.

`16 = t`

Check the solution is accurate.

`sqrt(16) = sqrt(16 - 12) + 2 -> 4 = 4 color(green)(√)`

Hopefully this helps!