Difference between revisions of "1950 AHSME Problems/Problem 38"
m (→Solution) |
|||
(One intermediate revision by one other user not shown) | |||
Line 8: | Line 8: | ||
By <math> | By <math> | ||
+ | |||
+ | Note: Alternatively, one may note that the equation is quadratic with a nonzero discriminant, so it will be satisfied for exactly two values of <math>x</math>. | ||
==See Also== | ==See Also== | ||
Line 15: | Line 17: | ||
[[Category:Introductory Algebra Problems]] | [[Category:Introductory Algebra Problems]] | ||
+ | {{MAA Notice}} |
Latest revision as of 00:35, 20 January 2024
Problem
If the expression has the value for all values of and , then the equation :
Solution
By , we have . Subtracting from both sides, giving . This factors to . Thus, , so the equation is .
Note: Alternatively, one may note that the equation is quadratic with a nonzero discriminant, so it will be satisfied for exactly two values of .
See Also
1950 AHSC (Problems • Answer Key • Resources) | ||
Preceded by Problem 37 |
Followed by Problem 39 | |
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.