Two positive numbers differ by 2. Their reciprocals add to 3. Find the numbers. (keep your answer in simplest radical form)
LOL - Yes, the reciprocals actually do add to 3
what are the two answers then
To solve this problem, let's assume the larger number is x and the smaller number is y. We are given two pieces of information:
1) "Two positive numbers differ by 2" means that x - y = 2.
2) "Their reciprocals add to 3" means that 1/x + 1/y = 3.
Now, let's solve these equations simultaneously to find the values of x and y:
From the first equation, we can rewrite it as x = y + 2.
Substituting this value of x into the second equation, we get 1/(y + 2) + 1/y = 3.
To simplify this equation, we need to find a common denominator:
- Multiply the first fraction by y/y to get y/(y(y + 2)), and
- Multiply the second fraction by (y + 2)/(y + 2) to get (y + 2)/(y(y + 2)).
Now, the equation becomes: y/(y(y + 2)) + (y + 2)/(y(y + 2)) = 3.
Combining the numerators, we have (y + y + 2)/(y(y + 2)) = 3.
Simplifying further, we have (2y + 2)/(y(y + 2)) = 3.
Now, cross multiply:
2y + 2 = 3y(y + 2).
Expanding the right side of the equation, we get:
2y + 2 = 3y^2 + 6y.
Move all terms to one side to get a quadratic equation:
3y^2 + (6y - 2y) - 2 = 0.
Simplifying, we have:
3y^2 + 4y - 2 = 0.
To solve this quadratic equation, we can use the quadratic formula:
y = (-b ± √(b^2 - 4ac)) / (2a).
In this case, a = 3, b = 4, and c = -2.
Plugging these values into the formula, we get:
y = (-4 ± √(4^2 - 4(3)(-2))) / (2(3)).
Simplifying further, we have:
y = (-4 ± √(16 + 24)) / 6.
y = (-4 ± √40) / 6.
y = (-4 ± 2√10) / 6.
Now, let's simplify the expression:
y = (-2/3) ± (√10)/3.
Since we are looking for positive numbers, we take the positive square root:
y = (-2/3) + (√10)/3.
Now, substitute this value of y back into the equation x = y + 2 to find x:
x = (-2/3) + (√10)/3 + 2.
x = (4/3) + (√10)/3.
Therefore, the two positive numbers that satisfy the given conditions are:
x = (4/3) + (√10)/3,
y = (-2/3) + (√10)/3.
n and n+2
1/n + 1/(n+2) = 3
(n+2) + n = 3 n(n+2) = 3 n^2 + 6 n
2 n + 2 = 3 n^2 + 6 n
3 n^2 + 4 n - 2 = 0
n = [ -4 +/- sqrt (16 + 24) ]/6
n = [ -4 +/- 2 sqrt (10) ]/6
n = -2/3 +/- (1/3) sqrt 10
says positive so use + sign
(sqrt 10 -2)/3
and
(sqrt 10 -2)/3 + 6/3 = (4+sqrt 10)/3