Difference between revisions of "1955 AHSME Problems/Problem 30"
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<math> \textbf{(A)}\ \text{two integral roots}\qquad\textbf{(B)}\ \text{no root greater than 3}\qquad\textbf{(C)}\ \text{no root zero}\\ \textbf{(D)}\ \text{only one root}\qquad\textbf{(E)}\ \text{one negative root and one positive root} </math> | <math> \textbf{(A)}\ \text{two integral roots}\qquad\textbf{(B)}\ \text{no root greater than 3}\qquad\textbf{(C)}\ \text{no root zero}\\ \textbf{(D)}\ \text{only one root}\qquad\textbf{(E)}\ \text{one negative root and one positive root} </math> | ||
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==Solution== | ==Solution== | ||
Since the question asks us about the unifying characteristic of all three equations' roots, we have to first determine them. | Since the question asks us about the unifying characteristic of all three equations' roots, we have to first determine them. | ||
Latest revision as of 20:10, 28 September 2024
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
In order to go back to the 1955 AHSME, click here.
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. 
