Chapter 7.7, Problem 27PPS

To determine

To Solve:

  $\sqrt{x-15}=3-\sqrt{x}$

Answer to Problem 27PPS

No real solutions.

Explanation of Solution

Given Information:

Given equation:

  $\sqrt{x-15}=3-\sqrt{x}$

Concept Used :

Solve Radical equations:

• Isolate the radical on one side of the equation.
• Raise each side of the equation to a power equal to the index of the radical to eliminate the radical .
• Solve the resulting polynomial equation.Check your results.
• When solving radical equations, the result may be a number that does not satisfy the original equation . Such a number is called an extraneous solution.

Calculation:

Given $\sqrt{x-15}=3-\sqrt{x}$

  $\begin{array}{l}\sqrt{x-15}=3-\sqrt{x}\\ \Rightarrow {\left(\sqrt{x-15}\right)}^2={\left(3-\sqrt{x}\right)}^2\mathrm{................................}[\text{Squaring each side]}\\ \Rightarrow x-15=9+x-6\sqrt{x}\mathrm{.................}[Simplify]\\ \Rightarrow -15=9-6\sqrt{x}\mathrm{..........................}[\text{Subtract x from each side]}\\ \Rightarrow -15-9=-6\sqrt{x}\mathrm{........................}[\text{Subtract 9 from each side]}\\ \Rightarrow 24=6\sqrt{x}\mathrm{.................................}[\text{Multiply -1 each side]}\\ \Rightarrow \frac{6\sqrt{x}}{6}=\frac{24}{6}\mathrm{...............................}[\text{Divide each side by 6]}\\ \Rightarrow \sqrt{x}=4\\ \Rightarrow x=\mathrm{16.....................................}[\text{Square each side]}\end{array}$

Check :

  

So, from the graph , there is no real solution to the given equation.

  $\begin{array}{l}\sqrt{x-15}=3-\sqrt{x}\\ \text{Substituting x=16 in the above equation:}\\ \text{Left hand side=}\sqrt{x-15}=\sqrt{16-15}=1\\ \text{Right hand side=}3-\sqrt{x}=3-\sqrt{16}=3-4=-1\\ 1\ne -1\\ \text{Left hand side }\ne \text{ Right hand side}\end{array}$