Difference between revisions of "1955 AHSME Problems/Problem 30"
Angrybird029 (talk | contribs) (Created page with "== Problem 30== Each of the equations <math>3x^2-2=25, (2x-1)^2=(x-1)^2, \sqrt{x^2-7}=\sqrt{x-1}</math> has: <math> \textbf{(A)}\ \text{two integral roots}\qquad\textbf{(B)...") |
Angrybird029 (talk | contribs) (→Solution) |
||
| Line 16: | Line 16: | ||
We can clearly see that, between all of the equations, there is <math>\boxed{\textbf{(B)} \text{no root greater than 3}}</math>. | We can clearly see that, between all of the equations, there is <math>\boxed{\textbf{(B)} \text{no root greater than 3}}</math>. | ||
| + | |||
| + | Note: There are probably extraneous solutions somewhere, but that does not affect the solution. | ||
| + | |||
==See Also== | ==See Also== | ||
Revision as of 16:05, 11 August 2020
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. 
