


Surprise! An Algeprize!...But Why?
Complete these steps:
 Start with the equation x^{2}+x6 = 0.
 On a grid, locate the points A(0,1) and B(1,6)...where the second point B represents negative of the parameter 1 of the xterm and the constant term 6.
 Draw the segment AB, locate its midpoint P, and draw a circle with center P and radius PA.
 Determine the coordinates of the two points where the circle intersects the Xaxis.
 Finally, factor the expression x^{2}+x6 to find the equation's roots.
 If done correctly, you should notice something quite interesting?
NOTE: You can explore this problem using the free DESMOS software suggested in the website for this week.
Does the process generalize? Complete these steps:
 Start with the equation x^{2}+px+q = 0.
 On a grid, locate the points A(0,1) and B(p,q)...where the second point B represents the negative of the parameter p of the xterm and the constant term q (signs are important!).
 Draw the segment AB, locate its midpoint P, and draw a circle with center P and radius PA.
 Determine the coordinates of the two points where the circle intersects the Xaxis....but what if the circle intersects the Xaxis in only one point...or no points?
 Finally, use the quadratic formula to determine the equation's roots.
 If done correctly, you again should notice something quite interesting?
Will this always work?
Why does it work?
Could you generalize it to a cubic equation?
Source: Adapted from H. Rice & R. Knight's Technical Mathematics With Calculus, 1966, p. 327
Hint: No hint needed...just try to follow the steps carefully.
Solution Commentary: Surprise: It always works!
But, any idea as to why?

