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)
What does the discriminant look like?
It looks like .. a number
5,2, 0 -1 - this is the discriminant for 4 different equations
What is the discriminant anyway?
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 $$a \ne 0 $$)
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 does the graph of a quadratic equation look like:
What is the solution of a quadratic equation:
The solution can be thought of in two different ways.
Algebraically, the solution occurs when y = 0. So the solution is where $$y =\color{Magenta}ax^2 + \red bx + \color{Blue}{c} $$ becomes $$0 =\color{Magenta}ax^2 + \red bx + \color{Blue}{c} $$
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?
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 SolutionsIf the discriminant is a perfect square the roots are rational. Otherwise, they are irrational. | |
Positive and a Perfect Square
b² − 4ac = perfect square |
Two Real Rational Solutions. | |
Positive and a not a perfect square
b² − 4ac = not a perfect square |
Two Real Irrational Solutions. | |
Discriminant is Zero
b² − 4ac = 0 |
One Real Solution | |
Negative Discriminant
b² − 4ac < 0 |
No Real Solutions Two Imaginary Solutions |