Difference between revisions of "1962 AHSME Problems/Problem 34"
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==Solution== | ==Solution== | ||
− | {{ | + | Factoring, we get |
+ | <cmath>x = K^2x^2 - 3K^2x + 2K^2</cmath> | ||
+ | <cmath>K^2x^2 - (3K^2+1)x + 2K^2 = 0</cmath> | ||
+ | At this point, we could use the quadratic formula, but we're only interested in whether the roots are real, which occurs if and only if the discriminant (<math>b^2-4ac</math>) is non-negative. | ||
+ | <cmath>(-3K^2-1)^2-8K^4\ge0</cmath> | ||
+ | <cmath>9K^4+6K^2+1-8K^4\ge0</cmath> | ||
+ | <cmath>K^4+6K^2+1\ge0</cmath> | ||
+ | This function of <math>K</math> is symmetric with respect to the y-axis. Plugging in <math>0</math> for <math>K</math> gives us <math>1</math>, and it is clear that the function is strictly increasing over <math>(0,\infty)</math>, and so it is positive for all real values of <math>K</math>. <math>\boxed{\textbf{(E)}}</math> |
Latest revision as of 17:37, 17 April 2014
Problem
For what real values of does
have real roots?
Solution
Factoring, we get
At this point, we could use the quadratic formula, but we're only interested in whether the roots are real, which occurs if and only if the discriminant (
) is non-negative.
This function of
is symmetric with respect to the y-axis. Plugging in
for
gives us
, and it is clear that the function is strictly increasing over
, and so it is positive for all real values of
.