Chapter 12, Problem 4CR

To determine

To define: The correct option of expression $c^2-10c-20=0$ ,solve by completing the square and the options are,

  $\begin{array}{l}\text{a}.\left\{-5+3\sqrt{5},-5-3\sqrt{5}\right\}\\ \text{b}\text{.}\left\{3\sqrt{5}+5,3\sqrt{5}-5\right\}\\ \text{c}.\left\{5+3\sqrt{5},5-3\sqrt{5}\right\}\\ \text{d}\text{.}\text{}\left\{-3\sqrt{5}+5,-3\sqrt{5}-5\right\}\end{array}$

Answer to Problem 4CR

The correct option of expression $c^2-10c-20=0$ is $\text{c}.\left\{5+3\sqrt{5},5-3\sqrt{5}\right\}$ .

Explanation of Solution

Given information:

The expression $c^2-10c-20=0$ and provided options are,

  $\begin{array}{l}\text{a}.\left\{-5+3\sqrt{5},-5-3\sqrt{5}\right\}\\ \text{b}\text{.}\left\{3\sqrt{5}+5,3\sqrt{5}-5\right\}\\ \text{c}.\left\{5+3\sqrt{5},5-3\sqrt{5}\right\}\\ \text{d}\text{.}\text{}\left\{-3\sqrt{5}+5,-3\sqrt{5}-5\right\}\end{array}$

The provided expression is $c^2-10c-20=0$ now solve for the value of $c$ 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 $c^2-10c-20=0$ is $\frac{-10}{2}=-5$

By step 2, square the result of step 1 is 25.

Add 20 on the both sides of the equation as,

  $\begin{array}{c}{c}^2-10c-20+20=20\\ c^2-10c=20\end{array}$

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

  $\begin{array}{l}{c}^2-10c+25=20+25\\ c^2-10c+25=45\end{array}$

By step 4 compete the square $c^2-10c+25={\left(c-5\right)}^2$

Put in the equation,

  ${\left(c-5\right)}^2=45$

Take under root on both sides of equation,

  $\begin{array}{c}\sqrt{{\left(c-5\right)}^2}=\pm \sqrt{45}\\ c-5=\pm 3\sqrt{5}\end{array}$

Furthermore,

  $\begin{array}{l}c=\pm 3\sqrt{5}+5\\ c=+3\sqrt{5}+5\text{or}c=-3\sqrt{5}+5\end{array}$

The value of $c$ is $\left\{+3\sqrt{5}+5,-3\sqrt{5}+5\right\}$ .

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