Chapter 16.3, Problem 7MRE

To determine

To solve: the given equation.

Answer to Problem 7MRE

The solution of the equation is $x=3$ .

Explanation of Solution

Given information:

An equation is given as

  $\sqrt{2x+3}=x$

Calculation:

Consider the given equation.

  $\sqrt{2x+3}=x$

Take square on both sides.

  $\begin{array}{c}{\left(\sqrt{2x+3}\right)}^2=x^2\\ 2x+3=x^2\\ x^2-2x-3=0\end{array}$

Now, solve the above quadratic equation by middle term factorization.

  $\begin{array}{l}{x}^2-2x-3=0\\ x^2-3x+x-3=0\\ x\left(x-3\right)+1\left(x-3\right)=0\\ \left(x+1\right)\left(x-3\right)=0\end{array}$

Therefore, $x+1=0$ or $x-3=0$ .

Thus, $x=-1$ or $x=3$

Now, check the solution is correct or not.

  $\begin{array}{l}\sqrt{2\left(-1\right)+3}=\left(-1\right)\\ \sqrt{-2+3}=\left(-1\right)\\ \sqrt{1}=\left(-1\right)\\ 1=-1\end{array}$

Which is false

And

  $\begin{array}{l}\sqrt{2\left(3\right)+3}=\left(3\right)\\ \sqrt{6+3}=\left(3\right)\\ \sqrt{9}=\left(3\right)\\ 3=3\end{array}$

Which is true.

Therefore, the solution of the equation is $x=3$ .