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Difference between revisions of "2014 AMC 12A Problems/Problem 19"

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==Solution 2==
 
==Solution 2==
 
Solve for <math>k</math> so <cmath>k=-\frac{12}{x}-5x.</cmath>  Note that <math>x</math> can be any integer in the range <math>[-39,0)\cup(0,39]</math>  so <math>k</math> is rational with <math>\lvert k\rvert<200</math>.  Hence, there are <math>39+39=\boxed{\textbf{(E) } 78}.</math>
 
Solve for <math>k</math> so <cmath>k=-\frac{12}{x}-5x.</cmath>  Note that <math>x</math> can be any integer in the range <math>[-39,0)\cup(0,39]</math>  so <math>k</math> is rational with <math>\lvert k\rvert<200</math>.  Hence, there are <math>39+39=\boxed{\textbf{(E) } 78}.</math>
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==Solution 3==
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Plug in k=200 to find the upper limit. You will find the limit to be a number from 0<x<-1 and one that is just below -39. All the integer values from -1 to -39 can be attainable through some value of k. Since the questions asks for the absolute value of k, we see that the answer is 39*2 = 78
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iron
  
 
==See Also==
 
==See Also==
 
{{AMC12 box|year=2014|ab=A|num-b=18|num-a=20}}
 
{{AMC12 box|year=2014|ab=A|num-b=18|num-a=20}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 20:04, 12 February 2019

Problem

There are exactly $N$ distinct rational numbers $k$ such that $|k|<200$ and \[5x^2+kx+12=0\] has at least one integer solution for $x$. What is $N$?

$\textbf{(A) }6\qquad \textbf{(B) }12\qquad \textbf{(C) }24\qquad \textbf{(D) }48\qquad \textbf{(E) }78\qquad$

Solution 1

Factor the quadratic into \[\left(5x + \frac{12}{n}\right)\left(x + n\right) = 0\] where $-n$ is our integer solution. Then, \[k = \frac{12}{n} + 5n,\] which takes rational values between $-200$ and $200$ when $|n| \leq 39$, excluding $n = 0$. This leads to an answer of $2 \cdot 39 = \boxed{\textbf{(E) } 78}$.

Solution 2

Solve for $k$ so \[k=-\frac{12}{x}-5x.\] Note that $x$ can be any integer in the range $[-39,0)\cup(0,39]$ so $k$ is rational with $\lvert k\rvert<200$. Hence, there are $39+39=\boxed{\textbf{(E) } 78}.$

Solution 3

Plug in k=200 to find the upper limit. You will find the limit to be a number from 0<x<-1 and one that is just below -39. All the integer values from -1 to -39 can be attainable through some value of k. Since the questions asks for the absolute value of k, we see that the answer is 39*2 = 78

iron

See Also

2014 AMC 12A (Problems • Answer Key • Resources)
Preceded by
Problem 18 		Followed by
Problem 20
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
All AMC 12 Problems and Solutions

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