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The Quadratic Formula and Discriminant
The solutions of the quadratic equation are given by the
Quadratic Formula :  Note: The quadratic formula is obtained by completing the square and solving for x. Try it! ☺ 
Example:
For the quadratic equation , we have a = 1, b
= 10, and c = 21, so
and 7
is called the discriminant because it discriminates among kinds of solutions. 
If a, b, and c are integers, then
If Discriminant  Then ( Kind of Solutions ) 
= 0  One rational solution 
> 0 and the square of an integer  Two rational solutions 
> 0 and not the square of an integer  Two irrational solutions 
< 0  Two complex (imaginary) solutions 
Examples:
1. has exactly one rational solution
because the discriminant = 0
(Show that x = 5 is the solution .)
2. has exactly two rational solutions
because the discriminant > 0
and the square of an integer .
( Show that 1 and 3/2 are the solutions .)
3. has exactly two irrational solutions
because the discriminant > 0 and
the square of an integer.
(Show that are
the solutions.)
4. has exactly two complex solutions
because the discriminant < 0.
(Show that
are the solutions.)
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