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Parabolic discovery (Posted on 2013-03-25) Difficulty: 3 of 5
Take two quadratic functions whose graphs are congruent parabolas such that one opens up, the other down and they share a common vertex not on the x-axis.

Clearly one has two real roots, and the other two complex roots.

Discover a relationship between these roots.

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Solution answer Comment 1 of 1

The real component of each of the complex roots is the mean between the two real roots. The imaginary components are plus or minus i times half the difference between the two real roots.

The set of four roots altogether are the north, south, east and west compass points on a circle centered on the real axis of the Argand plane.

  Posted by Charlie on 2013-03-25 16:53:46
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