Chapter 7.1, Problem 22WE

To determine

To Solve:

  $3n^2+12n+1=0$ by completing the square.

Answer to Problem 22WE

  $n=-2\pm \frac{\sqrt{33}}{3}$

Explanation of Solution

Given:

The equation is :

  $3n^2+12n+1=0$

Calculation:

Given:

  $\begin{array}{l}3n^2+12n+1=0\\ \Rightarrow 3n^2+12n=-\mathrm{1................}[subtract\text{ 1 from each side to isolate the constant]}\\ \Rightarrow \frac{3n^2+12n}{3}=-\frac{1}{3}\mathrm{..............}[{\text{divide each side by 3 to make the coefficient of n}}^2\text{ as 1]}\\ \Rightarrow n^2+4n=-\frac{1}{3}\\ \Rightarrow n^2+4n+{\left(\frac{4}{2}\right)}^2=-\frac{1}{3}+{\left(\frac{4}{2}\right)}^2\mathrm{....}\left[\begin{array}{l}\text{add the square of the half of the coefficient}\\ \text{of x each side}\end{array}\right]\\ \Rightarrow n^2+4n+4=-\frac{1}{3}+4\\ \Rightarrow {\left(n+2\right)}^2=\frac{11}{3}\mathrm{.................................}[{\left(a+b\right)}^2=a^2+2ab+b^2]\\ \Rightarrow n+2=\pm \sqrt{\frac{11}{3}}\mathrm{.................................}[\text{take square root each side]}\\ \Rightarrow n=-2\pm \sqrt{\frac{11}{3}}\mathrm{...................................}[subtract\text{ 2 from each side]}\\ \Rightarrow n=-2\pm \frac{\sqrt{11}}{\sqrt{3}}\cdot \frac{\sqrt{3}}{\sqrt{3}}\mathrm{...........................}[rationalize\text{ the denominator]}\\ \Rightarrow n=-2\pm \frac{\sqrt{33}}{3}\end{array}$

So,

  $n=-2\pm \frac{\sqrt{33}}{3}$