Solution assignment 01 Root equations

Solve:

$\sqrt{x+2}=x-1$

The following conditions have to be satisfied (because of the square roots):

$x\geq-2$ and $x\geq1$

so:

$x\geq1$

Squaring both sides yields the equation:

$x+2=(x-1)^2$

$x^2-3x-1=0$

This quadratic equation has the following solutions (by applying the $abc$ -formula):

$x_{1,2}=\displaystyle\frac{3\pm\sqrt{13}}{2}$

or:

$x_1\approx{3.30}$ of $x_2\approx{-0.30}$

The first equation satisfies the condition above, the second does not.

The solution:

$x=\displaystyle\frac{3+\sqrt{13}}{2}$

satisfies the original equation.