The Discriminant in Quadratic Equation
Nature of Quadratic Equation's roots, solutions

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:
What does the graph of a quadratic equation look like:
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 xaxis, the solution is where the parabola intercepts the xaxis. (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
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.


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


 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
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
In this quadratic equation,
y = 1x² − 2x + 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
In this quadratic equation,
y = x² − x − 2 and its solution
Discriminant
$$
\color{Red}{b^2}  4\color{Magenta}{a}\color{Yellow}{c}
\\
\color{Red}{(1)^2}  4\color{Magenta}{(1)}\color{Yellow}{(2)}
\\
1  8 = 1 + 8 = 9
$$
Problem 3)
Calculate the discriminant to determine the nature and number of solutions:
y = x² − 1
In this quadratic equation,
y = 1x² − 1
$$
\color{Red}{b^2}  4\color{Magenta}{a}\color{Yellow}{c}
\\
\color{Red}{(0)^2}  4\color{Magenta}{(1)}\color{Yellow}{(1)} = 4
$$
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
In this quadratic equation,
y = x² + 4x − 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
In this quadratic equation,
y = x²  4x + 5
$$
\color{Red}{b^2}  4\color{Magenta}{a}\color{Yellow}{c}
\\
\color{Red}{(4)^2}  4\color{Magenta}{(1)}\color{Yellow}{(5)}
\\
= 16  20 = 4
$$
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
y = x² + 4
$$
\color{Red}{b^2}  4\color{Magenta}{a}\color{Yellow}{c}
\\
\color{Red}{(0)^2}  4\color{Magenta}{(1)}\color{Yellow}{(4)} = 16
$$
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
y = x² + 25
$$
\color{Red}{b^2}  4\color{Magenta}{a}\color{Yellow}{c}
\\
\color{Red}{(0)^2}  4\color{Magenta}{(1)}\color{Yellow}{(25)} = 100
$$
The solutions are 5i and 5i
