Chapter 10.5, Problem 4MRE

To determine

To Solve:

  $\sqrt{x}+2=x$

Answer to Problem 4MRE

  $x=4$

Explanation of Solution

Given:

The equation is :

  $\sqrt{x}+2=x$

Calculation:

Substitute $\sqrt{x}=z$ in the given equation $\sqrt{x}+2=x$

  $\begin{array}{l}\Rightarrow z+2=z^2\\ \Rightarrow z^2-z-2=\mathrm{0........}[\text{subtract (z+2) from each side]}\\ \Rightarrow z^2-2z+z-2=\mathrm{0...}[\text{factor the middle term]}\\ \Rightarrow z(z-2)+1(z-2)=0\\ \Rightarrow (z-2)(z+1)=0\\ \Rightarrow z=2\text{ or }z=-\mathrm{1.......}[ab=0\iff a=0\text{ or }b=0]\end{array}$

Now, substitute

  $\begin{array}{l}z=\sqrt{x}\\ \Rightarrow \sqrt{x}=2\text{ or }\sqrt{x}=-1\\ \Rightarrow \sqrt{x}=2\text{ or no solution}\mathrm{..........}\text{[since the range of a square root function is always non-negative]}\\ \Rightarrow x=2^2\mathrm{......................................}[\text{square each side]}\\ \Rightarrow x=4\end{array}$

So

  $x=4$ is the solution.