Solve Quadratic Equations Using the Quadratic Formula: example 10.36

To determine the number of solutions of each quadratic equation, we will look at its discriminant.

1. ⓐ
   $\begin{array}{cccc}& & & \hfill 2v^2-3v+6=0\\ \text{The equation is in standard form, identify}a,b,c.\hfill & & & \hfill a=2,b=\mathrm{-3},c=6\\ \text{Write the discriminant.}\hfill & & & \hfill b^2-4ac\\ \text{Substitute in the values of}a,b,c.\hfill & & & \hfill {\left(3\right)}^2-4·2·6\\ \text{Simplify.}\hfill & & & \hfill \begin{array}{c}\hfill 9-48\\ \hfill \mathrm{-39}\end{array}\\ \text{Because the discriminant is negative, there are no real}\hfill & & & \\ \text{solutions to the equation.}\hfill & & & \end{array}$
   
2. ⓑ
   $\begin{array}{cccc}& & & \hfill 3x^2+7x-9=0\\ \text{The equation is in standard form, identify}a,b,c.\hfill & & & \hfill a=3,b=7,c=\mathrm{-9}\\ \text{Write the discriminant.}\hfill & & & \hfill b^2-4ac\\ \text{Substitute in the values of}a,b,c.\hfill & & & \hfill {\left(7\right)}^2-4·3·\left(\mathrm{-9}\right)\\ \text{Simplify.}\hfill & & & \hfill \begin{array}{c}\hfill 49+108\\ \hfill 157\end{array}\\ \text{Because the discriminant is positive, there are two}\hfill & & & \\ \text{solutions to the equation.}\hfill & & & \end{array}$
   
3. ⓒ
   $\begin{array}{cccc}& & & \hfill 5n^2+n+4=0\\ \text{The equation is in standard form, identify}a,b,\text{and}c.\hfill & & & \hfill a=5,b=1,c=4\\ \text{Write the discriminant.}\hfill & & & \hfill b^2-4ac\\ \text{Substitute in the values of}a,b,c.\hfill & & & \hfill {\left(1\right)}^2-4·5·4\\ \text{Simplify.}\hfill & & & \hfill \begin{array}{c}\hfill 1-80\\ \hfill \mathrm{-79}\end{array}\\ \text{Because the discriminant is negative, there are no real}\hfill & & & \\ \text{solutions to the equation.}\hfill & & & \end{array}$
   
4. ⓓ
   $\begin{array}{cccc}& & & \hfill 9y^2-6y+1=0\\ \text{The equation is in standard form, identify}a,b,c.\hfill & & & \hfill a=9,b=\mathrm{-6},c=1\\ \text{Write the discriminant.}\hfill & & & \hfill b^2-4ac\\ \text{Substitute in the values of}a,b,c.\hfill & & & \hfill {\left(\mathrm{-6}\right)}^2-4·9·1\\ \text{Simplify.}\hfill & & & \hfill \begin{array}{c}\hfill 36-36\\ \hfill 0\end{array}\\ \text{Because the discriminant is 0, there is one solution to the equation.}\hfill & & & \end{array}$