Chapter 12.2, Problem 5WE

To determine

To Solve: The equation $v^2-20v+19=0$ by completing the square.

Answer to Problem 5WE

The required solution set is $\left\{19,1\right\}$

Explanation of Solution

Given information:

  $v^2-20v+19=0$

Calculation:

  $\begin{array}{l}{v}^2-20v+19=0\\ v^2-20v=-19\end{array}$

(Subtract 19 from both sides)

  $v^2-20v+{\left(-\frac{20}{2}\right)}^2=-19+{\left(-\frac{20}{2}\right)}^2$

(Add square of half the coefficient of v to both sides)

  $\begin{array}{l}{v}^2-20v+{\left(-10\right)}^2=-19+{\left(-10\right)}^2\\ v^2-20v+100=-19+100\\ {\left(v-10\right)}^2=81\end{array}$

(Simplify)

  $\begin{array}{l}v-10=\sqrt{81}\\ v-10=\pm 9\end{array}$

(Take square root on both sides)

  $v=10\pm 9$

(Add 10 to each side)

  $\begin{array}{l}v=10+9\\ v=19\end{array}$

Or

  $\begin{array}{l}v=10-9\\ v=1\end{array}$

Therefore, the solution set is $\left\{19,1\right\}$ .