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The Discriminant in Quadratic Equation

Nature of Quadratic Equation's roots, solutions

Use The Quadratic Formula Calculator to see Quadratic Formula and discriminant in Action! This calculator will solve any quadratic equation you type in (even if solutions are imaginary)

Discriminant Worksheet

(free 25 question pdf with answer key)
Interactive Parabola Grapher
applet
Discriminant Calculator

What is the discriminant anyway?

Answer : The discriminant is a number that can be calculated from any quadratic equation
A quadratic equation is an equation that can be written as
    ax ² + bx + c where a ≠ 0

What is the formula for the Discriminant?

The discriminant in a quadratic equation is found by the following formula and the discriminant provides critical information regarding the nature of the roots/solutions of any quadratic equation.
discriminant= b² − 4ac

Example of the discriminant
  • Quadratic equation = y = 3x² + 9x + 5
  • The discriminant = 9 ² − 4 • 3 •5


Important pre requisites
To understand what the discriminant does, it's important that you have a good understanding of
What a quadratic equation is:
Answer: A quadratic equation is an equation in the form of $$y = ax^2 + bx + c$$ where $$a ne 0$$ Read more here on this topic
What does the graph of a quadratic equation look like:
Answer: A parabola.
What is the solution of a quadratic equation:
Answer: The solution can be thought of in two different ways.
Algebraically, the solution occurs when y = 0 . So the solution is where $$y = ax^2 + bx + c$$ becomes $$9 = ax^2 + bx + c$$
Graphically, since y = 0 is the x-axis, the solution is where the parabola intercepts the x-axis. (This only works for real solutions).
In the picture below, the left parabola has 2 real solutions (red dots), the middle parabola has 1 real solution (red dot) and the right most parabola has no real solutions (yes, it does have imaginary ones)


What does this formula tell us?

Answer: The discriminant tells us the following information about a quadratic equation:
  • If the solution is a real number or an imaginary number.
  • If the solution is rational or if it is irrational.
  • If the solution is 1 unique number or two different numbers
Nature of the Solutions
Value of the discriminant Type and number of Solutions Example of graph
Positive Discriminant

b² − 4ac > 0
Two Real Solutions
If the discriminant is a perfect square the roots are rational. Otherwise, they are irrational.
picture of positive discriminant
Positive and a Perfect Square

b² − 4ac = perfect square
Two Real Rational Solutions. picture of positive discriminant
Positive and a not a perfect square

b² − 4ac = not a perfect square
Two Real Irrational Solutions. picture of positive discriminant
Discriminant is Zero

b² − 4ac = 0
One Real Solution
Negative Discriminant

b² − 4ac < 0
No Real Solutions
Two Imaginary Solutions
picture of imaginary solutions
  • Example 1
      Quadratic Equation: y = x² + 2x + 1
    • a = 1
    • b = 2
    • c = 1
    The discriminant for this equation is
    $$ \color{Red}{b^2} - 4\color{Magenta}{a}\color{Yellow}{c} \\ \color{Red}{2^2} - 4\color{Magenta}{(1)}\color{Yellow}{(1)} = 0 $$
    Since the discriminant is zero, there should be 1 real solution to this equation. Below is a picture representing the graph and one solution of this quadratic equation
    Graph of y = x² + 2x + 1
    Picture of graph of  solved quadratic formula
Practice Problems


Problem 1) Calculate the discriminant to determine the number and nature of the solutions of the following quadratic equation:
y = x² − 2x + 1
Answer


Problem 2) Use the discriminant to find out the nature and number of solutions:
y = x² − x − 2
Answer


Problem 3) Calculate the discriminant to determine the nature and number of solutions:
y = x² − 1
Answer


Problem 4) Calculate the discriminant to determine the nature and number of solutions:
y = x² + 4x − 5
Answer


Problem 5) Calculate the discriminant to determine the nature and number of solutions:
y = x² - 4x + 5
Answer


Problem 6) Find the discriminant to determine the nature and number of solutions:
y = x² + 4
Answer


Problem 7) Find the discriminant to determine the nature and number of solutions:
y = x² + 25
Answer
Discriminant Worksheet

(free 25 question pdf with answer key)
Interactive Parabola Grapher
applet
Discriminant Calculator