1964 AHSME Problems/Problem 30
Contents
Problem
The larger root minus the smaller root of the equation is
Solution 1
Dividing the quadratic by to obtain a monic polynomial will give a linear coefficient of . Rationalizing the denominator gives:
Dividing the constant term by (and using the same radical conjugate as above) gives:
So, dividing the original quadratic by the coefficient of gives
From the quadratic formula, the positive difference of the roots is . Plugging in gives:
Note that if we take of one of the answer choices and square it, we should get . The only answers that are (sort of) divisible by are , so those would make a good first guess. And given that there is a negative sign underneath the radical, is the most logical place to start.
Since of the answer is , and , the answer is indeed .
Solution 2
Submitted by BinouTheGuineaPig | A step-by-step solution
The original equation can be manipulated as follows.
Substituting ,
or
First root of :
Second root of :
Now, to find which root of is larger:
Assume that
, and so
which is true. Hence, .
Finally, finding the difference between the larger and smaller roots of :
Therefore, the answer is .
See Also
1964 AHSC (Problems • Answer Key • Resources) | ||
Preceded by Problem 29 |
Followed by Problem 31 | |
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