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)

What does the discriminant look like?

Answer

It looks like .. a number

5,2, 0 -1 - this is the discriminant for 4 different equations

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?

Answer

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:

Answer

A parabola

What is the solution of a quadratic equation:

Answer

The solution can be thought of in two different ways.

  1. Algebraically, the solution occurs when y = 0. So the solution is where $$y = ax^2 + bx + c$$ becomes $$9 = ax^2 + bx + c$$

  2. 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 SolutionsIf the discriminant is a perfect square the roots are rational. Otherwise, they are irrational. picture of positive discriminant
Positive and a Perfect Square

² − 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

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

Practice 1

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

In this quadratic equation,y = 1 − 2x + 1

  • a = 1
  • b = − 2
  • c = 1

Using our general formula, the discriminant is

$$ \color{Red}{b^2} - 4\color{Magenta}{a}\color{Yellow}{c} \\ \color{Red}{(-2)^2} - 4\color{Magenta}{(1)}\color{Yellow}{(1)} = 4 $$

Since the discriminant is zero, we should expect 1 real solution which you can see pictured in the graph below.

Practice 2

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

In this quadratic equation, y = x² − x − 2 and its solution

  • a = 1
  • b = − 1
  • c = − 2

Discriminant

$$ \color{Red}{b^2} - 4\color{Magenta}{a}\color{Yellow}{c} \\ \color{Red}{(-1)^2} - 4\color{Magenta}{(1)}\color{Yellow}{(-2)} \\ 1 - -8 = 1 + 8 = 9 $$

Since the discriminant is positive and rational , there should be 2 real rational solutions to this equation. As you can see below, if you use the quadratic formula to find the actual solutions, you do indeed get 2 real rational solutions.

Practice 3

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

In this quadratic equation, y = 1− 1

  • a = 1
  • b = 0
  • c = − 1

$$ \color{Red}{b^2} - 4\color{Magenta}{a}\color{Yellow}{c} \\ \color{Red}{(0)^2} - 4\color{Magenta}{(1)}\color{Yellow}{(-1)} = 4 $$

Since the discriminant is positive and a perfect square, we have two real solutions that are rational.

Again if you'd like to see the actual solutions and the graph, just look below:

Practice 4

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

In this quadratic equation, y = x² + 4x − 5

  • a = 1
  • b = 4
  • c = − 5

$$ \color{Red}{b^2} - 4\color{Magenta}{a}\color{Yellow}{c} \\ \color{Red}{(4)^2} - 4\color{Magenta}{(1)}\color{Yellow}{(-5)} \\ 16 - 4(-5) = 16 +20 \\ = 36 $$

Since this quadratic equation's discriminant is positive and a perfect square, there are two real solutions that are rational.

Practice 5

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

In this quadratic equation, y = x² - 4x + 5

  • a = 1
  • b = -4
  • c = 5

$$ \color{Red}{b^2} - 4\color{Magenta}{a}\color{Yellow}{c} \\ \color{Red}{(-4)^2} - 4\color{Magenta}{(1)}\color{Yellow}{(5)} \\ = 16 - 20 = -4 $$

Since the discriminant is negative , there are no real solutions to this quadratic equation. The only solutions are imaginary.

Below is a picture of this quadratic's graph

Practice 6

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

y = x² + 4

  • a = 1
  • b = 0
  • c = 4

$$ \color{Red}{b^2} - 4\color{Magenta}{a}\color{Yellow}{c} \\ \color{Red}{(0)^2} - 4\color{Magenta}{(1)}\color{Yellow}{(4)} = -16 $$

Since the discriminant is negative , there are two imaginary solutions to this quadratic equation.

The solutions are 2i and -2i

Below is a a picture of this equations graph

Practice 7

Find the discriminant to determine the nature and number of solutions: y = x² + 25

y = x² + 25

  • a = 1
  • b = 0
  • c = 25

$$ \color{Red}{b^2} - 4\color{Magenta}{a}\color{Yellow}{c} \\ \color{Red}{(0)^2} - 4\color{Magenta}{(1)}\color{Yellow}{(25)} = -100 $$

Since the discriminant is negative , there are two imaginary solutions to this quadratic equation.

The solutions are 5i and -5i