Chapter 6.5, Problem 23WE

a.

To determine

Solve the radical equation $x=3+2\sqrt{x}$ by method of squaring both sides.

Answer to Problem 23WE

The solution for the radical equation is $x=9$ .

Explanation of Solution

Given information: The given radical equation is,

  $x=3+2\sqrt{x}\mathrm{....}\left(1\right)$

Calculation: Simplify (1)

  $\begin{array}{c}x=3+2\sqrt{x}\mathrm{......}\left(\text{Given}\right)\\ \text{or, }x-3=2\sqrt{x}\mathrm{.....}\left(\text{Re-arranged}\right)\\ x^2-6x+9=4x\mathrm{....}\left(\text{Square}\right)\\ \text{or, }{x}^2-10x+9=0\end{array}$

Further solving,

  $\begin{array}{c}\\ \left(x-1\right)\left(x-9\right)=0\\ x-1=0\text{or}x-9=0\\ x=1\text{ or}x=9\end{array}$

Since $x=1$ does not satisfy (1) and $x=9$ satisfy (1). Therefore $x=9$ is solution for the given radical equation.

b.

To determine

Solve the linear equation $x=3+x\sqrt{2}$ without using square-method.

Answer to Problem 23WE

The solution for the linear equation is $x=-3-3\sqrt{2}$ .

Explanation of Solution

Given information: The given linear equation is,

  $x=3+x\sqrt{2}\mathrm{....}\left(2\right)$

Calculation: Simplify (2)

  $\begin{array}{c}x=3+x\sqrt{2}\mathrm{......}\left(\text{Given}\right)\\ \text{or, }\left(1-\sqrt{2}\right)x=3\\ x=\frac{3}{1-\sqrt{2}}\\ x=\left(\frac{3}{1-\sqrt{2}}\right)\left(\frac{1+\sqrt{2}}{1+\sqrt{2}}\right)\mathrm{......}\left(\text{Rationalized}\right)\end{array}$

Further solving,

  $\begin{array}{c}x=\frac{3+3\sqrt{2}}{1-2}\\ x=\frac{3+3\sqrt{2}}{-1}\\ x=-3-3\sqrt{2}\end{array}$