Perfect Square: Solving Two EquationsDate: 6/21/96 at 20:58:1 From: Sheldon Wilson Subject: Find x so that Two Expressions are Perfect Squares Find the value of x such that x^2 + 5 is a perfect square AND x^2 - 5 is also a perfect square. The value of x must not contain surds; that is, x must equal M / N where M = {1 or 2 or 3 or 4 or . . .} and N = {1 or 2 or 3 or . . .} I have solved this problem correctly but I am not satisfied with my solution. I am looking for a more elegant solution. Thanks. Date: 6/26/96 at 11:5:38 From: Doctor Brian Subject: Re: Find x so that Two Expressions are Perfect Squares Try looking at some of the differences between perfect squares. You're trying to find a difference of ten (between x^2 - 5 and x^2 + 5) 1-0=1 4-0=4 4-1=3 9-0=9 9-1=8 9-4=5 16-0=16 16-1=15 16-4=12 16-9=7 25-0=25 25-1=24 25-4=21 25-9=16 25-16=9 At this point, the difference between *consecutive* perfect squares is more than ten: (x + 1)^2 - x^2 = x^2 + 2x + 1 - x^2 = 2x + 1, which for x>5 gives an answer greater than or equal to 11. Finally, since squaring a number is an increasing function, then any larger gap between the numbers must give a larger difference that between consecutive numbers. For instance, the gap between the squares of 6 and 8 must be larger than the difference in the squares of 6 and 7. So the only possible way to give a difference less than 10 between the numbers would be to use numbers less than five. But the above table shows that there is no pair of numbers less than five whose squares give a difference of ten. -Doctor Brian, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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