Difference between revisions of "1965 AHSME Problems/Problem 1"
(→Solution) |
Hastapasta (talk | contribs) |
||
Line 6: | Line 6: | ||
− | ==Solution== | + | ==Solution 1== |
Solution by e_power_pi_times_i | Solution by e_power_pi_times_i | ||
Take the logarithm with a base of <math>2</math> to both sides, resulting in the equation <math>2x^2-7x+5 = 0</math>. Factoring results in <math>(2x-5)(x-1) = 0</math>, so there are <math>\boxed{\textbf{(C) } 2}</math> real solutions. | Take the logarithm with a base of <math>2</math> to both sides, resulting in the equation <math>2x^2-7x+5 = 0</math>. Factoring results in <math>(2x-5)(x-1) = 0</math>, so there are <math>\boxed{\textbf{(C) } 2}</math> real solutions. | ||
+ | |||
+ | ==Solution2== | ||
+ | Notice that <math>a^0=1, a>0</math>. So <math>2^0=1</math>. So <math>2x^2-7x+5=0</math>. Evaluating the discriminant, we see that it is equal to <math>7^2-4*2*5=9</math>. So this means that the equation has two real solutions. Therefore, select <math>\boxed{B}</math>. | ||
+ | |||
+ | ~hastapasta | ||
+ | |||
+ | ==See Also== | ||
+ | {{AHSME box|year=1965|before=First Question|num-a=2}} | ||
+ | {{MAA Notice}} | ||
+ | [[Category:AHSME]][[Category:AHSME Problems]] |
Revision as of 19:30, 2 April 2022
Contents
Problem
The number of real values of satisfying the equation
is:
Solution 1
Solution by e_power_pi_times_i
Take the logarithm with a base of to both sides, resulting in the equation
. Factoring results in
, so there are
real solutions.
Solution2
Notice that . So
. So
. Evaluating the discriminant, we see that it is equal to
. So this means that the equation has two real solutions. Therefore, select
.
~hastapasta
See Also
1965 AHSME (Problems • Answer Key • Resources) | ||
Preceded by First Question |
Followed by Problem 2 | |
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.