Chapter 3.6, Problem 36E

Solution

To calculate: The value of x for the equation $x=\sqrt{16x+225}$ and also check the extraneous solution.

The resultant answer is $x=25$ and the extraneous solution is -9.

Given information:

The equation is $x=\sqrt{16x+225}$ .

Formula used:

Raise each side of the equation to the same power.

Calculation:

Write the original equation:

  $x=\sqrt{16x+225}$

Square each side of the equation,

  $\begin{array}{l}{x}^2={\left(\sqrt{16x+225}\right)}^2\\ x^2=16x+225\end{array}$

Write the equation in standard form and factor the equation,

  $\begin{array}{c}{x}^2-16x-225=0\\ x^2-25x+9x-225=0\\ x\left(x-25\right)+9\left(x-25\right)=0\\ \left(x-25\right)\left(x+9\right)=0\end{array}$

Now apply zero-product property,

  $\begin{array}{c}x-25=0\text{or}x+9=0\\ x=25\text{or}x=-9\end{array}$

Now check $x=25\text{or}x=-9$ in the original equation.

  $x=\sqrt{16x+225}$

Now substitute 25 for x and simplify,

  $\begin{array}{l}25\stackrel{?}{=}\sqrt{16\cdot 25+225}\\ 25\stackrel{?}{=}\sqrt{400+225}\\ 25\stackrel{?}{=}\sqrt{625}\\ 25=25\end{array}$

Now substitute -9 for x and simplify,

  $\begin{array}{l}-9\stackrel{?}{=}\sqrt{16\cdot \left(-9\right)+225}\\ -9\stackrel{?}{=}\sqrt{-144+225}\\ -9\stackrel{?}{=}\sqrt{81}\\ -9\ne 9\end{array}$

Therefore, the only solution is 25 and the apparent solution -9 is extraneous.