The Discriminant in Quadratic Equations--visual tutorial with examples, practice problems and free printable pdf

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)

Example 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

Problem 1) Calculate the discriminant to determine the number and nature of the solutions of the following quadratic equation:

y = x² − 2x + 1

Answer

In this quadratic equation,

y = 1x² − 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.

Problem 2) Use the discriminant to find out the nature and number of solutions:

y = x² − x − 2

Answer

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..

Problem 3) Calculate the discriminant to determine the nature and number of solutions:

y = x² − 1

Answer

In this quadratic equation,

y = 1x² − 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:

Problem 4) Calculate the discriminant to determine the nature and number of solutions:

y = x² + 4x − 5

Answer

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 .

Problem 5) Calculate the discriminant to determine the nature and number of solutions:

y = x² - 4x + 5

Answer

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

Problem 6) Find the discriminant to determine the nature and number of solutions:

y = x² + 4

Answer

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

Problem 7) Find the discriminant to determine the nature and number of solutions:

y = x² + 25

Answer

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

The Discriminant in Quadratic Equation

Nature of Quadratic Equation's roots, solutions

What is the discriminant anyway?

What is the formula for the Discriminant?

What does this formula tell us?