Chapter 12, Problem 3CR

To determine

To define: The correct option of expression $x^2-4x-77=0$ ,solve by completing the square and the options are,

  $\begin{array}{l}\text{a}.\left\{11,7\right\}\\ \text{b}\text{.}\left\{-11,-7\right\}\\ \text{c}.\left\{11,-7\right\}\\ \text{d}\text{.}\text{}\left\{-11,7\right\}\end{array}$

Answer to Problem 3CR

The correct option of expression $x^2-4x-77=0$ is $\text{c}.\left\{11,-7\right\}$ .

Explanation of Solution

Given information:

The expression $x^2-4x-77=0$ and provided options are,

  $\begin{array}{l}\text{a}.\left\{11,7\right\}\\ \text{b}\text{.}\left\{-11,-7\right\}\\ \text{c}.\left\{11,-7\right\}\\ \text{d}\text{.}\text{}\left\{-11,7\right\}\end{array}$

The provided expression is $x^2-4x-77=0$ now solve for the value of $x$ by the completing the square method.

Method of completing the square;

For $x^2+\text{b}x+\underset{\_}{?}$

Step 1: Find half the coefficient of $x$ : $\frac{\text{b}}{\text{2}}$

Step 2: Square the result of step 1: ${\left(\frac{\text{b}}{\text{2}}\right)}^2$

Step 3: Add the result of step 2 to $x^2+\text{b}x$ : $x^2+\text{b}x+{\left(\frac{\text{b}}{\text{2}}\right)}^2$

Step 4: Completed the square: $x^2+\text{b}x+{\left(\frac{\text{b}}{\text{2}}\right)}^2={\left(x+\frac{\text{b}}{\text{2}}\right)}^2$

Apply the method of completing square,

By step 1, half of the coefficient of $x$ in equation $x^2-4x-77=0$ is $\frac{-4}{2}=-2$

By step 2, square the result of step 1 is 4.

Add 77 on the both sides of the equation as,

  $\begin{array}{c}{x}^2-4x-77+77=77\\ x^2-4x=77\end{array}$

By step 3 add the result of step 2 to both sides of equation,

  $\begin{array}{l}{x}^2-4x+\left(4\right)=77+\left(4\right)\\ x^2-4x+\left(4\right)=81\end{array}$

By step 4 compete the square $x^2-4x+4={\left(x-2\right)}^2$

Put in the equation,

  ${\left(x-2\right)}^2=81$

Take under root on both sides of equation,

  $\begin{array}{c}\sqrt{{\left(x-2\right)}^2}=\pm \sqrt{81}\\ x-2=\pm 9\end{array}$

Furthermore,

  $\begin{array}{l}x=\pm 9+2\\ x=11\text{or}x=-7\end{array}$

The value of $x$ is $\left\{11,-7\right\}$ .

Thus, the option $\text{c}$ is correct.