SOLUTION: The sum of the recipricals of 2 consectutive even integars is 11/60. Find the integars.

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Question 256684: The sum of the recipricals of 2 consectutive even integars is 11/60. Find the integars.
Found 2 solutions by stanbon, vleith:
Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
The sum of the reciprocals of 2 consecutive even integers is 11/60. Find the integers.
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Let the integers be:
1st 2x
2nd 2x+2
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Equation:
1/(2x) + 1/(2x+2) = 11/60
Multiply thru by 2 to get:
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1/x + 1/(x+1) = 11/30
(x+1 + x)/(x(x+1)) = 11/30
(2x+1)/(x^2+x) = 11/30
Cross-multiply
60x + 30 = 11x^2 + 11x
11x^2 - 49x - 30 = 0
(x-5)(11x+6) = 0
Positive solution:
x = 5
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1st number: 2x = 10
2nd number: 2x+2 = 12
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Cheers,
Stan H.

Answer by vleith(2983) About Me  (Show Source):
You can put this solution on YOUR website!
let the two integers be represented by x and (x+2)
Then their reciprocals are 1%2Fx and 1%2F%28x%2B2%29
You are told the sum of the reciprocals is 11/60
1%2Fx+%2B+1%2F%28x%2B2%29+=++11%2F60
%28%28x%2B2%29+%2B+x%29%2F%28x%2A%28x%2B2%29%29+=+11%2F60
%282x%2B2%29%2F%28x%5E2%2B2x%29+=+11%2F60
120x%2B120+=+11x%5E2+%2B+22x
0+=+11x%5E2+-+98x+-120
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 11x%5E2%2B-98x%2B-120+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-98%29%5E2-4%2A11%2A-120=14884.

Discriminant d=14884 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--98%2B-sqrt%28+14884+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-98%29%2Bsqrt%28+14884+%29%29%2F2%5C11+=+10
x%5B2%5D+=+%28-%28-98%29-sqrt%28+14884+%29%29%2F2%5C11+=+-1.09090909090909

Quadratic expression 11x%5E2%2B-98x%2B-120 can be factored:
11x%5E2%2B-98x%2B-120+=+11%28x-10%29%2A%28x--1.09090909090909%29
Again, the answer is: 10, -1.09090909090909. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+11%2Ax%5E2%2B-98%2Ax%2B-120+%29


So one integer is 10 and the next one is 12
Check your answer. Does 1/10 + 1/12 = 11/60???