Difference between revisions of "1974 AHSME Problems/Problem 10"
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==Problem== | ==Problem== | ||
| − | What is the smallest integral value of <math> k </math> such that | + | <!-- don't remove the following tag, for PoTW on the Wiki front page--><onlyinclude>What is the smallest integral value of <math> k </math> such that |
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<cmath> 2x(kx-4)-x^2+6=0 </cmath> | <cmath> 2x(kx-4)-x^2+6=0 </cmath> | ||
| − | + | has no real roots?<!-- don't remove the following tag, for PoTW on the Wiki front page--></onlyinclude> | |
| − | has no real roots? | ||
<math> \mathrm{(A)\ } -1 \qquad \mathrm{(B) \ }2 \qquad \mathrm{(C) \ } 3 \qquad \mathrm{(D) \ } 4 \qquad \mathrm{(E) \ }5 </math> | <math> \mathrm{(A)\ } -1 \qquad \mathrm{(B) \ }2 \qquad \mathrm{(C) \ } 3 \qquad \mathrm{(D) \ } 4 \qquad \mathrm{(E) \ }5 </math> | ||
Latest revision as of 16:14, 18 November 2015
Problem
What is the smallest integral value of 
such that
![\[2x(kx-4)-x^2+6=0\]](http://latex.artofproblemsolving.com/6/8/4/684666741d25abc15ab55d259373ad3447b8bd32.png)
has no real roots?
Solution
Expanding, we have 
, or 
. For this quadratic not to have real roots, it must have a negative discriminant. Therefore, 
. From here, we can easily see that the smallest integral value of is 
.
See Also
| 1974 AHSME (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 | ||
| All AHSME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
