5.14: Using the Discriminant
The profit on your school fundraiser is represented by the quadratic expression
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Khan Academy: Discriminant for Types of Solutions for a Quadratic
Guidance
From the previous concept, the Quadratic Formula is
Investigation: Solving Equations with Different Types of Solutions
1. Solve
2. Solve
3. Solve
4. Look at the values of the discriminants from Steps 13. How do they differ? How does that affect the final answer?
From this investigation, we can conclude:
 If
b2−4ac>0 , then the equation has two real solutions.  If
b2−4ac=0 , then the equation has one real solution; a double root.  If
b2−4ac<0 , then the equation has two imaginary solutions.
Example A
Determine the type of solutions
Solution: Find the discriminant.
At this point, we know the answer is going to be negative, so there is no need to continue (unless we were solving the problem). This equation has two imaginary solutions.
Example B
Solve the equation from Example A to prove that it does have two imaginary solutions.
Solution: Use the Quadratic Formula.
Example C
Find the value of the determinant and state how many solutions the quadratic has.
Solution: Use the discriminant. a = 3, b = 5, and c = 12</math>
This quadratic has two real solutions.
Intro Problem Revisit Set the expression
At this point, we know the answer is zero, so the equation has one real solution. Therefore, there is one real breakeven point.
Guided Practice
1. Use the discriminant to determine the type of solutions
2. Use the discriminant to determine the type of solutions
3. Solve the equation from #1.
Answers
1.
This equation has two real solutions.
2.
This equation has one real solution.
3.
Vocabulary
 Discriminant

The value under the radical in the Quadratic Formula,
b2−4ac . The discriminant tells us number and type of solution(s) a quadratic equation has.
Practice
Determine the number and type of solutions each equation has.

x2−12x+36=0 
5x2−9=0 
2x2+6x+15=0 
−6x2+8x+21=0 
x2+15x+26=0 
4x2+x+1=0
Solve the following equations using the Quadratic Formula.

x2−17x−60=0 
6x2−20=0 
2x2+5x+11=0
Challenge Determine the values for

x2+2x+c=0 
x2−6x+c=0 
x2+12x+c=0  What is the discriminant of
x2+2kx+4=0 ? Write your answer in terms ofk .  For what values of
k will the equation have two real solutions?  For what values of
k will the equation have one real solution?  For what values of
k will the equation have two imaginary solutions?
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Here you'll use the discriminant of the Quadratic Formula to determine how many real solutions an equation has.