1989 AHSME Problems/Problem 28

Problem

Find the sum of the roots of $\tan^2x-9\tan x+1=0$ that are between $x=0$ and $x=2\pi$ radians.

$\mathrm{(A) \frac{\pi}{2} } \qquad \mathrm{(B) \pi } \qquad \mathrm{(C) \frac{3\pi}{2} } \qquad \mathrm{(D) 3\pi } \qquad \mathrm{(E) 4\pi }$

Solution

The roots of $t^2-9t+1=0$ are positive and distinct, so by considering the graph of $y=\tan x$ , the smallest two roots of the original equation $x_1,\ x_2$ are between $0$ and $\tfrac\pi{2}$ , and the two other roots are $\pi+x_1,\ \pi+x_2$ .

Then from the quadratic equation we discover that the product $\tan x_1\tan x_2=1$ which implies that $\tan(x_1+x_2)$ does not exist. The bounds then imply that $x_1+x_2=\tfrac\pi{2}$ . Thus $x_1+x_2+\pi+x_1+\pi+x_2=3\pi$ which is $\rm{(D)}$ .

1989 AHSME (Problems • Answer Key • Resources)
Preceded by Problem 27	Followed by Problem 29
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30
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1989 AHSME Problems/Problem 28

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