Chapter 7.2, Problem 2ST

To determine

To solve: The given equation by completing the square.

Answer to Problem 2ST

The solution of the equation is $y=-\frac{3}{2}\pm \frac{2\sqrt{57}}{12}$ .

Explanation of Solution

Given:

The equation is

  $3y^2+9y=-2$

Calculation:

Keep the constant -2 on the right side.

  $3y^2+9y=-2$

The coefficient of y² is not 1. Divide the whole equation by 3.

  $\begin{array}{l}\frac{3}{3}{y}^2+\frac{9}{3}y=\frac{-2}{3}\\ y^2+3y=\frac{-2}{3}\end{array}$

Calculate the half of the y term and square it.

  ${\left(3\times \frac{1}{2}\right)}^2=\frac{9}{4}$

Add this $\frac{9}{4}$ to both the sides of the equation.

  $y^2+3y+\frac{9}{4}=\frac{-2}{3}+\frac{9}{4}$

The left hand side of the equation can be written in whole square terms.

  ${\left(y+\frac{3}{2}\right)}^2=\frac{19}{12}$

Take the square root of the equation and solve for y .

  $\begin{array}{l}\left(y+\frac{3}{2}\right)=\pm \sqrt{\frac{19}{12}}\\ \left(y+\frac{3}{2}\right)=\pm \sqrt{\frac{19\times 12}{12\times 12}}\\ \left(y+\frac{3}{2}\right)=\pm \frac{2\sqrt{57}}{12}\end{array}$

Isolate y on left hand side.

  $y=-\frac{3}{2}\pm \frac{2\sqrt{57}}{12}$

Therefore, the solution of the equation is $y=-\frac{3}{2}\pm \frac{2\sqrt{57}}{12}$ .

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