Chapter 11, Problem 20GR

To determine

Solve the equation: $\sqrt{x}+2=x$

Answer to Problem 20GR

Solutions of the equation $\sqrt{x}+2=x$ are $x=4\text{and}x=1$

Explanation of Solution

Given:

The equation: $\sqrt{x}+2=x$

Concept Used:

To get rid from the square root sign, first square both side of the equation. Rule: ${(\sqrt{x})}^2=x$

Zero-Product Property: If we multiply two (or more) things together and the result is equal to zero, then we know that  at least one  of those things that we multiplied must also have been equal to zero.

Calculation:

  $\begin{array}{l}\sqrt{x}+2=x\\ \sqrt{x}+2-2=x-2[\text{Subtract}2\text{from}\text{both}\text{sides}]\\ \sqrt{x}=x-2\\ {(}^{}=(x-2){}^2\\ x=x^2-4x+4\\ x-x=x^2-4x-x+4[\text{Subtract}x\text{from}\text{both}\text{sides}]\\ x^2-4x-x+4=0\\ (x-4)(x-1)=0\end{array}$

Factor form: $x^2-4x-x+4=0\Rightarrow (x-4)(x-1)=0$

Now use zero product property:

  $\begin{array}{l}{x}^2-4x-x+4=0\\ (x-4)(x-1)=0\\ x-4=0\text{and}x-1=0[\text{Zero-Product Property}]\\ x=4\text{and}x=1\end{array}$

Thus, Solutions of the equation $\sqrt{x}+2=x$ are $x=4\text{and}x=1$