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Study Guides > Intermediate Algebra

Read: The Discriminant

Learning Objectives

  • Define the discriminant and use it to classify solutions to quadratic equations

The Discriminant

The quadratic formula not only generates the solutions to a quadratic equation, it tells us about the nature of the solutions. When we consider the discriminant, or the expression under the radical, b24ac{b}^{2}-4ac, it tells us whether the solutions are real numbers or complex numbers, and how many solutions of each type to expect. The table below relates the value of the discriminant to the solutions of a quadratic equation.
Value of Discriminant Results
b24ac=0{b}^{2}-4ac=0 One repeated rational solution
b24ac>0{b}^{2}-4ac>0, perfect square Two rational solutions
b24ac>0{b}^{2}-4ac>0, not a perfect square Two irrational solutions
b24ac<0{b}^{2}-4ac<0 Two complex solutions

A General Note: The Discriminant

For ax2+bx+c=0a{x}^{2}+bx+c=0, where aa, bb, and cc are real numbers, the discriminant is the expression under the radical in the quadratic formula: b24ac{b}^{2}-4ac. It tells us whether the solutions are real numbers or complex numbers and how many solutions of each type to expect.

Example

Use the discriminant to find the nature of the solutions to the following quadratic equations:
  1. x2+4x+4=0{x}^{2}+4x+4=0
  2. 8x2+14x+3=08{x}^{2}+14x+3=0
  3. 3x25x2=03{x}^{2}-5x - 2=0
  4. 3x210x+15=03{x}^{2}-10x+15=0

Answer: Calculate the discriminant b24ac{b}^{2}-4ac for each equation and state the expected type of solutions.

  1. x2+4x+4=0{x}^{2}+4x+4=0b24ac=(4)24(1)(4)=0{b}^{2}-4ac={\left(4\right)}^{2}-4\left(1\right)\left(4\right)=0. There will be one repeated rational solution.
  2. 8x2+14x+3=08{x}^{2}+14x+3=0b24ac=(14)24(8)(3)=100{b}^{2}-4ac={\left(14\right)}^{2}-4\left(8\right)\left(3\right)=100. As 100100 is a perfect square, there will be two rational solutions.
  3. 3x25x2=03{x}^{2}-5x - 2=0b24ac=(5)24(3)(2)=49{b}^{2}-4ac={\left(-5\right)}^{2}-4\left(3\right)\left(-2\right)=49. As 4949 is a perfect square, there will be two rational solutions.
  4. 3x210x+15=03{x}^{2}-10x+15=0b24ac=(10)24(3)(15)=80{b}^{2}-4ac={\left(-10\right)}^{2}-4\left(3\right)\left(15\right)=-80. There will be two complex solutions.

We have seen that a quadratic equation may have two real solutions, one real solution, or two complex solutions. In the Quadratic Formula, the expression underneath the radical symbol determines the number and type of solutions the formula will reveal. This expression, b24acb^{2}-4ac, is called the discriminant of the equation ax2+bx+c=0ax^{2}+bx+c=0. Let’s think about how the discriminant affects the evaluation of b24ac \sqrt{{{b}^{2}}-4ac}, and how it helps to determine the solution set.
  • If b24ac>0b^{2}-4ac>0, then the number underneath the radical will be a positive value. You can always find the square root of a positive, so evaluating the Quadratic Formula will result in two real solutions (one by adding the positive square root, and one by subtracting it).
  • If b24ac=0b^{2}-4ac=0, then you will be taking the square root of 00, which is 00. Since adding and subtracting 00 both give the same result, the "±\pm" portion of the formula doesn't matter. There will be one real repeated solution.
  • If b24ac<0b^{2}-4ac<0, then the number underneath the radical will be a negative value. Since you cannot find the square root of a negative number using real numbers, there are no real solutions. However, you can use imaginary numbers. You will then have two complex solutions, one by adding the imaginary square root and one by subtracting it.

Example

Use the discriminant to determine how many and what kind of solutions the quadratic equation x24x+10=0x^{2}-4x+10=0 has.

Answer: Evaluate b24acb^{2}-4ac. First note that a=1,b=4a=1,b=−4, and c=10c=10. b24ac(4)24(1)(10)\begin{array}{c}b^{2}-4ac\\\left(-4\right)^{2}-4\left(1\right)\left(10\right)\end{array} The result is a negative number. The discriminant is negative, so the quadratic equation has two complex solutions. 1640=2416–40=−24

Answer

The quadratic equation x24x+10=0x^{2}-4x+10=0 has two complex solutions.

In the last example, we will draw a correlation between the number and type of solutions to a quadratic equation and the graph of it's corresponding function.

Example

Use the following graphs of quadratic functions to determine how many and what type of solutions the corresponding quadratic equation f(x)=0f(x)=0 will have.  Determine whether the discriminant will be greater than, less than, or equal to zero for each. a. Screen Shot 2016-08-04 at 12.10.26 PM b. Screen Shot 2016-08-04 at 12.12.08 PM c. Screen Shot 2016-08-04 at 12.14.18 PM

Answer: a. This quadratic function does not touch or cross the x-axis, therefore the corresponding equation f(x)=0f(x)=0 will have complex solutions. This implies that b24ac<0b^{2}-4ac<0. b. This quadratic function touches the x-axis exactly once, which implies there is one repeated solution to the equation f(x)=0f(x)=0.  We can then say that b24ac=0b^{2}-4ac=0 c. In our final graph, the quadratic function crosses the x-axis twice which tells us that there are two real number solutions to the equation f(x)=0f(x)=0, and therefore b24ac>0b^{2}-4ac>0.

We can summarize our results as follows:
Discriminant Number and Type of Solutions Graph of Quadratic Function
b24ac<0b^{2}-4ac<0 two complex solutions will not cross the x-axis
b24ac=0b^{2}-4ac=0 one real repeated solution will touch x-axis once
b24ac>0b^{2}-4ac>0  two real solutions  will cross x-axis twice
In the following video we show more examples of how to use the discriminant to describe the type of solutions to a quadratic equation. https://youtu.be/hSWs0VUyn1k

Summary

The discriminant of the Quadratic Formula is the quantity under the radical, b24ac {{b}^{2}}-4ac. It determines the number and the type of solutions that a quadratic equation has. If the discriminant is positive, there are 22 real solutions. If it is 00, there is 11 real repeated solution. If the discriminant is negative, there are 22 complex solutions (but no real solutions). The discriminant can also tell us about the behavior of the graph of a quadratic function.

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