Difference between revisions of "1958 AHSME Problems/Problem 10"
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Revision as of 05:08, 3 October 2014
Problem
For what real values of 
, other than $k \equal{} 0$ (Error compiling LaTeX. Unknown error_msg), does the equation $x^2 \plus{} kx \plus{} k^2 \equal{} 0$ (Error compiling LaTeX. Unknown error_msg) have real roots?
Solution
An expression of the form 
has at least one real root when 
.
Substituting for 
and 
for 
, we have
![\[k^2-4k^2 \geq 0\]](http://latex.artofproblemsolving.com/8/b/a/8bafbc8b780624858927dfaa7bd0341b0059dcbf.png)
![\[-3k^2 \geq 0\]](http://latex.artofproblemsolving.com/6/7/7/677866618f6edd311316f2a60d47bb37e5309981.png)
but the range of 
is ![$(-\infty,0]$](http://latex.artofproblemsolving.com/5/0/d/50d25d3e486146edd225ea505aaa53016f41edb7.png)
, so the answer is 
See also
| 1958 AHSC (Problems • Answer Key • Resources) | ||
| Preceded by Problem 9 |
Followed by Problem 11 | |
| 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 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 • 41 • 42 • 43 • 44 • 45 • 46 • 47 • 48 • 49 • 50 | ||
| All AHSME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
