Chapter 1.6, Problem 38E

To determine

To calculate: The solution of the equation, $2x=\sqrt{-5x+24}-3$ and check the solution.

The solution of the given equation, $2x=\sqrt{-5x+24}-3$ is $x=\frac{3}{4}$.

Calculation:

Consider the provided equation,

$2x=\sqrt{-5x+24}-3$

Isolate the radical.

$2x+3=\sqrt{-5x+24}$

Square each side and solve.

$\begin{array}{c}{\left(2x+3\right)}^2=-5x+24\\ 9+4x^2+12x=-5x+24\end{array}$

Write the above equation in standard form.

$4x^2+17x-15=0$

Now factorize the above equation.

$\left(4x-3\right)\left(x+5\right)=0$

Put the first factor equal to zero.

$\begin{array}{c}4x-3=0\\ x=\frac{3}{4}\end{array}$

Put the second factor equal to zero.

$\begin{array}{c}x+5=0\\ x=-5\end{array}$

Check:

Put $x=\frac{3}{4},-5$ in the equation, $2x=\sqrt{-5x+24}-3$.

First put $x=\frac{3}{4}$.

$\begin{array}{l}2\left(\frac{3}{4}\right)\stackrel{?}{=}\sqrt{-5\left(\frac{3}{4}\right)+24}-3\\ \frac{3}{2}\stackrel{?}{=}\sqrt{\frac{-15}{4}+24}-3\\ \frac{3}{2}\stackrel{?}{=}\sqrt{\frac{81}{4}}-3\\ \frac{3}{2}=\frac{3}{2}\end{array}$

Which is true.

Now put $x=-5$.

$\begin{array}{c}2\left(-5\right)\stackrel{?}{=}\sqrt{-5\left(-5\right)+24}-3\\ -10\stackrel{?}{=}\sqrt{-25+24}-3\\ -10\stackrel{?}{=}\sqrt{-1}-3\end{array}$

Which is false, and therefore $x=-5$ is not the solution.

Hence, the solution of the given equation is $x=\frac{3}{4}$.