Chapter 12, Problem 4CT

To determine

To calculate: The value of $b$ from the equation $b^2-4b-10=0$

Answer to Problem 4CT

The values are $b\text{=}\sqrt{14}+2\text{or}\text{ b}\text{=}-\sqrt{14}-2$ .

Explanation of Solution

Given information:

The expression is given as:

  $b^2-4b-10=0$

Formula used:

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: $x^2+\text{b}x+{\left(\frac{\text{b}}{\text{2}}\right)}^2={\left(x+\frac{\text{b}}{\text{2}}\right)}^2$

Calculation:

Apply the method of completing square,

By step 1, half of the coefficient of $x$ in equation $b^2-4b-10=0$ is $\frac{-4}{2}=-2$

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

Add 10 on the both sides of the equation as,

  $\begin{array}{c}{b}^2-4b-10+10=10\\ b^2-4b=10\end{array}$

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

  $b^2-4b+4=14$

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

Put in the equation,

  ${\left(b\text{-}\text{2}\right)}^2=14$

Take under root on both sides of equation,

  $\sqrt{{\left(\text{b}-2\right)}^2}=\pm \sqrt{14}$

Furthermore,

  $\begin{array}{c}\left(\text{b}-2\right)=\pm \sqrt{14}\\ b=\pm \sqrt{14}+2\end{array}$

The value of $b\text{=}\sqrt{14}+2\text{or}\text{ b}\text{=}-\sqrt{14}-2$ is $$ .