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Discriminant

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Using the Discriminant

The profit on your school fundraiser is represented by the quadratic expression -5p^2 + 400p - 8000 , where p is your price point. How many real break-even points will you have?

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Khan Academy: Discriminant for Types of Solutions for a Quadratic

Guidance

From the previous concept, the Quadratic Formula is x=\frac{-b \pm \sqrt{b^2-4ac}}{2a} . The expression under the radical, b^2-4ac , is called the discriminant. You can use the discriminant to determine the number and type of solutions an equation has.

Investigation: Solving Equations with Different Types of Solutions

1. Solve x^2-8x-20=0 using the Quadratic Formula. What is the value of the discriminant?

x &=\frac{8 \pm \sqrt{{\color{red}144}}}{2}\\&=\frac{8 \pm 12}{2} \rightarrow 10, -2

2. Solve x^2-8x+6=0 using the Quadratic Formula. What is the value of the discriminant?

x &=\frac{8 \pm \sqrt{{\color{red}0}}}{2}\\&=\frac{8 \pm 0}{2} \rightarrow 4

3. Solve x^2-8x+20=0 using the Quadratic Formula. What is the value of the discriminant?

x &=\frac{8 \pm \sqrt{{\color{red}-16}}}{2}\\&=\frac{8 \pm 4i}{2} \rightarrow 4 \pm 2i

4. Look at the values of the discriminants from Steps 1-3. How do they differ? How does that affect the final answer?

From this investigation, we can conclude:

  • If b^2-4ac > 0 , then the equation has two real solutions.
  • If b^2-4ac = 0 , then the equation has one real solution; a double root.
  • If b^2-4ac < 0 , then the equation has two imaginary solutions.

Example A

Determine the type of solutions 4x^2-5x+17=0 has.

Solution: Find the discriminant.

b^2-4ac &=(-5)^2-4(4)(17)\\&=25-272

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.

x = \frac{5 \pm \sqrt{25-272}}{8}=\frac{5 \pm \sqrt{-247}}{8}= \frac{5}{8} \pm \frac{\sqrt{247}}{8}i

Example C

Find the value of the determinant and state how many solutions the quadratic has.

3x^2-5x-12=0

Solution: Use the discriminant. a = 3, b = -5, and c = -12</math>

 \sqrt{(-5)^2 - 4(3)(-12)}=\sqrt{25+144}=\sqrt{169}=13

This quadratic has two real solutions.

Intro Problem Revisit Set the expression -5p^2 + 400p - 8000 equal to zero and then find the discriminant.

-5p^2 + 400p - 8000 = 0

b^2-4ac &=(400)^2-4(-5)(-8000)\\&=160000-160000 = 0

At this point, we know the answer is zero, so the equation has one real solution. Therefore, there is one real break-even point.

Guided Practice

1. Use the discriminant to determine the type of solutions -3x^2-8x+16=0 has.

2. Use the discriminant to determine the type of solutions 25x^2-80x+64=0 has.

3. Solve the equation from #1.

Answers

1. b^2-4ac &=(-8)^2-4(-3)(16)\\&=64+192\\&=256

This equation has two real solutions.

2. b^2-4ac &=(-80)^2-4(25)(64)\\&=6400-6400\\&=0

This equation has one real solution.

3. x = \frac{8 \pm \sqrt{256}}{-6}=\frac{8 \pm 16}{-6}=-4, \frac{4}{3}

Vocabulary

Discriminant
The value under the radical in the Quadratic Formula, b^2-4ac . 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.

  1. x^2-12x+36=0
  2. 5x^2-9=0
  3. 2x^2+6x+15=0
  4. -6x^2+8x+21=0
  5. x^2+15x+26=0
  6. 4x^2+x+1=0

Solve the following equations using the Quadratic Formula.

  1. x^2-17x-60=0
  2. 6x^2-20=0
  3. 2x^2+5x+11=0

Challenge Determine the values for c that make the equation have a) two real solutions, b) one real solution, and c) two imaginary solutions.

  1. x^2+2x+c=0
  2. x^2-6x+c=0
  3. x^2+12x+c=0
  4. What is the discriminant of x^2+2kx+4=0 ? Write your answer in terms of k .
  5. For what values of k will the equation have two real solutions?
  6. For what values of k will the equation have one real solution?
  7. For what values of k will the equation have two imaginary solutions?

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