When using the quadratic formula to solve a quadratic 
equation , the discriminant is 
. This discriminant can be positive, 
zero, or negative. (When the discriminant is negative, 
then we have the square root of a negative number. 
This is called an imaginary number, sqrt(-1) = i. )

Explain what the value of the discriminant means to 
the graph of y = ax2 + bx + c. Hint: Chose values of 
a, b and c to create a particular discriminant.  Then, 
graph the corresponding equation?

Choose a = 1, b = 4, c = -21

Then the equation is y = x² + 4x - 21 and the 
discriminant is
(4)² - 4(1)(-21) = 16 + 84 = 100, which is positive.  
The graph intersects the x-axis twice, once at 
x = -7 and again at x = 3. There are two real 
zeros, -7, and 3



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Now choose a = 1, b = 4, c = 4

Then the equation is y = x² + 4x + 4 and the 
discriminant is (4)² - 4(1)(4) = 16 - 16 = 0.  
The graph just touches the x-axis at -2.  
There is just one real zeros, -2.  [This zero 
is said to have multiplicity 2 because people 
like to think of the graph as "crossing the 
x-axis twice at the same point", and "both its 
two zeros are the same, i.e., 'merging' into
one".]



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Finally choose a = 1, b = 4, c = 6

Then the equation is y = x² + 4x + 6 and the 
discriminant is (4)² - 4(1)(6) = 16 - 24 = -8, 
which is negative.  The graph does not cross 
or touch the x-axis.  Therefore it has no real
zeros, which means that both its solutions are 
imaginary.
 


Edwin