Quadratic Equations
Using the Discriminant to predict the roots of a quadratic
equation
The discriminant of a quadratic equation is the value under
the square root sign in the quadratic formula.
Remember the quadratic formula for an equation in the form ax
+ bx + c = 0 is:
From this formula the discriminant is: b
 4ac
When you evaluate the discriminant for a quadratic equation,
if the result is:
positive 
You will have 2 different real solutions
to the equation 

If this number is a perfect square
number, there will be 2 different rational answers. If
this number is a not perfect square number, there will be
2 different irrational answers. 
zero 
You will have 1 real, rational solution
to the equation  that is, there will be a repeated
answer 
negative 
You will have no real solutions to the
equation (only imaginary answers) 
Examples:
Use the discriminant to predict the roots of the following
equations:
1. x + 7x + 12 = 0 
a = 1 
b = 7 
c = 12 
b  4ac = 7  4(1)(12) = 49  48 = 1 
Since the result is positive, there should be 2 different real
solutions.
In fact, there will be 2 different rational solutions because
1 is a perfect square number.
(Perfect square numbers are: 1, 4, 9, 16, 25, 36, 49, 64, 81,
100, 121, 144, etc)
2. x + 7x + 3 = 0 
a = 1 
b = 7 
c = 3 
b  4ac = 7  4(1)(3) = 49  12 = 37 
Since the result is positive, there should be 2 different real
solutions.
In fact, there will be 2 different irrational solutions
because 37 is not a perfect square number.
3. x + 4x + 4 = 0 
a = 1 
b = 4 
c = 4 
b  4ac = 4  4(1)(4) = 16  16 = 0 
Since the result is zero, there should be only one real,
rational solution
4. x  x + 4 = 0 
a = 1 
b = 1 
c = 4 
b  4ac = (1)  4(1)(4) = 1  16 = 15 
Since the result is negative, there should be no real
solutions.
