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) About Me  (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) About Me  (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.