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· See the graphs below for quadratic equations based on their discriminant values.

Comments and Conclusions?

quadratic equations and their solutions

x2 - 2x - 3 = 0

x2 - 6x + 9 = 0

x2 + 3x + 3 = 0

http://www.purplemath.com/modules/complex/complx33.gif

http://www.purplemath.com/modules/complex/complx34.gif

http://www.purplemath.com/modules/complex/complx35.gif

a positive number inside the square root

zero inside the square root

a negative number inside the square root

two distinct real-number solutions

one (repeated) real-number solution

no real-number solutions

associated functions and their graphs

y = x2 - 2x - 3

y = x2 - 6x + 9

y = x2 + 3x + 3

http://www.purplemath.com/modules/complex/complx27.gif

http://www.purplemath.com/modules/complex/complx28.gif

http://www.purplemath.com/modules/complex/complx29.gif

two distinct x-intercepts

one (repeated) x-intercept

no x-intercepts

What is the point of the above table? The point is that there is a clear and consistent connection between solutions of quadratic equations (where you've got "(quadratic) = 0") and the graphs of the associated functions (which will be "y = (quadratic)"); namely, that the real solutions of the equation will be the x-intercepts of the graph.