10.3 Solve quadratic equations using the quadratic formula  (Page 3/5)

Solution

Multiply both sides by the LCD, 6, to clear the fractions.
Multiply.
Subtract 2 to get the equation in standard form.
Identify the a, b, c values.
Write the Quadratic Formula.
Then substitute in the values of a, b, c.
Simplify.
Simplify the radical.
Factor out the common factor in the numerator.
Remove the common factors.
Rewrite to show two solutions.
Check. We leave the check to you.

Solution

Add 25 to get the equation in standard form.
Identify the a, b, c values.
Write the Quadratic Formula.
Then substitute in the values of a, b, c.
Simplify.
Simplify the radical.
Simplify the fraction.
Check. We leave the check to you.

Did you recognize that $4x^2-20x+25$ is a perfect square?

Solution

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. ⓓ
   $\begin{array}{cccc}& & & \hfill 9y^2-6y+1=0\\ \text{The equation is in standard form, identify}a,b,c.\hfill & & & \hfill a=9,b=\mathrm{-6},c=1\\ \text{Write the discriminant.}\hfill & & & \hfill b^2-4ac\\ \text{Substitute in the values of}a,b,c.\hfill & & & \hfill {\left(\mathrm{-6}\right)}^2-4·9·1\\ \text{Simplify.}\hfill & & & \hfill \begin{array}{c}\hfill 36-36\\ \hfill 0\end{array}\\ \text{Because the discriminant is 0, there is one solution to the equation.}\hfill & & & \end{array}$