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

1 Answer
Apr 19, 2016

Answer:

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!