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 19:10, 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.
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
