1955 AHSME Problems/Problem 30
Problem 30
Each of the equations has:
Solution
Since the question asks us about the unifying characteristic of all three equations' roots, we have to first determine them.
can be rewritten as
, which gives the following roots
and
.
can be expanded to
, which in turn leads to
. The roots here are
and
.
, when squared, also turns into a quadratic equation:
. Binomial factoring gives us the roots
and
.
We can clearly see that, between all of the equations, there is .
Note: There are probably extraneous solutions somewhere, but that does not affect the solution.
See Also
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These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions.