2021 JMPSC Invitationals Problems/Problem 1
Problem
The equation where
is some constant, has
as a solution. What is the other solution?
Solution
Since must be a solution,
must be true. Therefore,
. We plug this back into the original quadratic to get
. We can solve this quadratic to get
. We are asked to find the 2nd solution so our answer is
~Grisham
Solution 2
Plug to get
, so
, or
, meaning the other solution is
~Geometry285
Solution 3
Plugging in
, we get
, therefore,
Finally, we get the other root is
.
- kante314 -
Solution 4
We can rearrange the equation to get that . Then, by Vieta's Formulas, we have
and
where
is the second root of the quadratic. Solving for
tells us that the answer is
.
~Mathdreams
See also
- Other 2021 JMPSC Invitationals Problems
- 2021 JMPSC Invitationals Answer Key
- All JMPSC Problems and Solutions
The problems on this page are copyrighted by the Junior Mathematicians' Problem Solving Competition.