# 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.