The Discriminant
In the quadratic formula, the radicand, b^{2}  4ac, is called the discriminant
of the quadratic equation ax^{2} + bx + c = 0.
We can use the discriminant to determine the nature of the solutions of a
quadratic equation without having to solve the equation.
Discriminant 
Solutions 

b^{2}  4ac > 0 
two different real numbers 

b^{2}  4ac = 0 
two identical real numbers 

b^{2}  4ac < 0 
no real number solutions 

If the discriminant is a perfect square, the
solutions will not only be real numbers,
they will also be rational numbers.
Example
Use the discriminant to determine the nature of the solutions of this
quadratic equation: 2x^{2}  x + 7 = 0
Solution
The equation has the form ax^{2} + bx + c = 0 where a
= 2, b = 1, and c = 7. 

Substitute the values of a, b, and c into the discriminant and simplify. 
b^{2}  4ac 
= (1)^{2}  4(2)(7)
= 1 + 56
= 57 
The discriminant is 57, a positive number.
So the equation 2x^{2}  x + 7 = 0 has two unequal real number solutions.
