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C-104 Appendix C<br />
MINI PROJECT<br />
A Geometric Interpretation of Complex Roots<br />
As you know, if a quadratic equation f(x) x 2 Bx C 0 has two real<br />
roots, there is a geometric interpretation: The roots are the x-intercepts for the<br />
graph of f. Less well known is the fact that there is a geometric interpretation<br />
when the quadratic equation has nonreal complex roots. If these roots are<br />
a bi, where a and b are real numbers, then the coordinates of the vertex of<br />
the graph of f are (a, b 2 ).<br />
(a) Check that this is true in the case of the quadratic equation<br />
f(x) x 2 6x 25 0.<br />
(b) Show that the result holds in general. That is, assume that the roots of<br />
f(x) x 2 Bx C 0 are a bi, where a and b are real numbers with<br />
b 0. Show that the vertex of the graph of f is (a, b 2 ). Hint: Use the theorem<br />
on sums and products of the roots of a quadratic equation to express<br />
A and B in terms of a and b.
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