Question

How would you simplify `sqrt(x+1)=3- sqrt( x-2)`?

Answer 1

Make a simplifying substitution, like `y^2 = x+1`, then solve to find `x=3`

Explanation:

One way to solve this type of equation is by doing substitutions to simplify one of the square-roots. In this case, it would be really nice if we could do the square root of `x+1` on the left hand side, so let's make a substitution that allows that to happen, i.e. let

`y^2 = x+1`

From which we can solve for `x`, so that we can substitute that expression into the rest of the equation:

`x=y^2-1`

Substituting these into the equation we get:

`y=3-sqrt(y^2-3)`

let's take the `3` to the other side (subtract 3 from both sides) and then square both sides and then simplify:

`(y-3)^2 = y^2-3`

`y^2-6y+9 = y^2-3`

`-6y-12 = 0`

`-6y=-12`

`y=2`

Finally, we can substitute this into our original expression for `x` in terms of `y`

`x=2^2-1 = 3`

Let's check this answer by substituting it into the original equation:

`sqrt(3+1) = 3-sqrt(3-2)`

`2=2` correct!