Chapter 12, Problem 24CCSR

To determine

Find the correct option.

Answer to Problem 24CCSR

  $H$

Explanation of Solution

Given information:

Simplify,

  $x+2=\sqrt{2x^2-x-2}$

Options:-

  $\begin{array}{l}F 6,1\\ G -6,-1\\ H 6,-1\\ I -6,1 \end{array}$

Calculation:

Use the identity

  ${\left(a+b\right)}^2=a^2+2ab+b^2$

Consider the given equation

  $x+2=\sqrt{2x^2-x-2}$

Squaring both sides:

  ${\left(x+2\right)}^2=2x^2+x+2$

Now, applying the identity:

  $x^2+4x+4=2x^2-x-2$

Now, taking the terms to one side:

  $2x^2-x-2-x^2-4x-4=0$

Take the like terms together:

  $\left(2x^2-x^2\right)+\left(-x-4x\right)+\left(-2-4\right)=0$

Further solving:

  $x^2-5x-6=0$

Now, use the middle term splitting to get the value of $x$ :

  $\begin{array}{l}{x}^2+x-6x-6=0\\ x\left(x+1\right)-6\left(x+1\right)=0\\ \left(x+1\right)\left(x-6\right)=0\end{array}$

That is,

  $\begin{array}{l}\left(x+1\right)=0or\left(x-6\right)=0\\ x=-1orx=6\end{array}$

Hence, the solution is $6,-1$ .

Hence, the option $H$ is correct.