SOLUTION: The difference between two positive numbers is 4√3. The product of the two
numbers is 8. What is the absolute value of the difference of their reciprocals?
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Question 258748: The difference between two positive numbers is 4√3. The product of the two
numbers is 8. What is the absolute value of the difference of their reciprocals?
Found 2 solutions by edjones, Theo:
Answer by edjones(8007) (Show Source): You can put this solution on YOUR website!
x-y=4sqrt(3)
xy=8
.
y=8/x
x-8/x=4sqrt(3)
x^2-8=4xsqrt(3)
x^2-4xsqrt(3)-8=0
x^2-4xsqrt(3)+12=8+12 complete the square.
|x-2sqrt(3)|=+-2sqrt(5) Take sqrt of each side.
x=2sqrt(3)+-2sqrt(5)
.
Ed
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
let a be the larger of the numbers.
let b be the smaller of the numbers.
you have:
a-b = 4*sqrt(3)
you also have:
a*b = 8
the question is:
What is the absolute value of the difference of their reciprocals?
their reciprocals are:
(1/a) for a and (1/b) for b
since a > b, then 1/b > 1/a
this is derived as follows:
a > b
divide both sides of the equation by ab to get:
1/b > 1/a
an example would be 3 > 2.
divide both sides of this equation by 6 to get:
1/2 > 1/3
the absolute value of the difference of their reciprocals is:
|1/b - 1/a|
looking at the expression within the absolute value signs, we see:
(1/b - 1/a)
if we multiply this by (ab)/(ab), we get:
((a-b)/ab)
|1/b - 1/a| is therefore equal to |(a-b)/(ab)|
we know that a-b = 4*sqrt(3).
we also know that ab = 8
|1/b - 1/a| is therefore equal to |4*sqrt(3)/8|
this simplifies to |sqrt(3)/2|
since sqrt(3)/2 is always positive, then we have:
|1/b - 1/a| is equal to |sqrt(3)/2|
since sqrt(3)/2 is always positive, we have:
|1/b - 1/a| = |sqrt(3)/2| = sqrt(3)/2
this is because, by definition:
|x| = x if x is positive, and |x| = (-x) if x is negative.
your answer is:
the absolute value of the difference of their reciprocals equals sqrt(3)/2.
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