Question

In the following exercises, solve by using the Quadratic Formula.

$3y^2+5y-2=0$

Short answer

The solutions of the given equation are $y=\frac{1}{3}$and $y=-2$.

Step-by-step solution

Step 1. Given information.

The given equation is:

$3y^2+5y-2=0$

Step 2. Substitute the values in Quadratic Formula.

Quadratic Formula for the equation $a{x}^2+bx+c=0$is:

$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

In the given equation, $a=3,b=5,c=-2$. Substitute these values in Quadratic Formula.

$y=\frac{-5\pm \sqrt{5^2-4\left(3\right)(-2)}}{2\left(3\right)}$

Step 3. Simplify the value.

On simplification, we get

$\begin{array}{rcl}y& =& \frac{-5\pm \sqrt{25+24}}{6}\\ y& =& \frac{-5\pm \sqrt{49}}{6}\\ y& =& \frac{-5\pm 7}{6}\end{array}$

Now,

$\begin{array}{rcl}y& =& \frac{-5+7}{6}\\ y& =& \frac{2}{6}\\ y& =& \frac{1}{3}\end{array}$

And,

$\begin{array}{rcl}y& =& \frac{-5-7}{6}\\ y& =& \frac{-12}{6}\\ y& =& -2\end{array}$

Step 4. Conclusion.

The solutions of the given equation are $y=\frac{1}{3}$and $y=-2$.