1989 AHSME Problems/Problem 28
Problem
Find the sum of the roots of that are between and radians.
Solution
The roots of are positive and distinct, so by considering the graph of , the smallest two roots of the original equation are between and , and the two other roots are .
Then from the quadratic equation we discover that the product which implies that does not exist. The bounds then imply that . Thus which is .
See also
1989 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 27 |
Followed by Problem 29 | |
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The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.
tan^2(x) -9tan(x)+1 We treat tan(x1) and tan(x2) as the roots of our equation Because tan(x1) * tan(x2) = 1 by Vieta's formula, x1 + x2 = 0.5pi. Because the principle values of x1 and x2 are acute and our range for x is [0,2pi], we have four values of x that satisfy the quadratic: x1, x2, x1+pi, x2+pi Summing these, we obtain 2(x1+x2) + 2pi. Using the fact that x1+x2=0.5pi 2(0.5pi) + 2pi = 3pi