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Surprise! An Alge-prize!...But Why?

Complete these steps:

• 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 x-term 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 X-axis.
• Finally, factor the expression x2+x-6 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:

• 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 x-term 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 X-axis....but what if the circle intersects the X-axis 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?