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

Apr 19, 2016

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:

${\left(y - 3\right)}^{2} = {y}^{2} - 3$

${y}^{2} - 6 y + 9 = {y}^{2} - 3$

$- 6 y - 12 = 0$

$- 6 y = - 12$

$y = 2$

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!