Chapter 12.5, Problem 12MRE

To determine

To Solve: The equation $12y^2+24y=5$ and to write the irrational roots in the simplest form.

Answer to Problem 12MRE

The required answer is $\left\{-1+\frac{\sqrt{17}}{2\sqrt{3}},-1-\frac{\sqrt{17}}{2\sqrt{3}}\right\}$ .

Explanation of Solution

Given information:

The equation $12y^2+24y=5$

Calculation:

  $\begin{array}{l}12y^2+24y=5\\ y^2+2y=\frac{5}{12}\end{array}$

(Divide each side by 12)

  $y^2+2y+{\left(1\right)}^2=\frac{5}{12}+{\left(1\right)}^2$

(Add Square of half the coefficient of y)

  $\begin{array}{l}{\left(y+1\right)}^2=\frac{17}{12}\\ y+1=\pm \sqrt{\frac{17}{12}}\\ y+1=\pm \frac{\sqrt{17}}{2\sqrt{3}}\end{array}$

(Simplify)

  $y=-1\pm \frac{\sqrt{17}}{2\sqrt{3}}$

So,

  $y=-1+\frac{\sqrt{17}}{2\sqrt{3}}$

Or

  $y=-1-\frac{\sqrt{17}}{2\sqrt{3}}$

Therefore, the solution set is $\left\{-1+\frac{\sqrt{17}}{2\sqrt{3}},-1-\frac{\sqrt{17}}{2\sqrt{3}}\right\}$ .