SOLUTION: The sum of the reciprocals of two consectutive intergers is 3/2.What are the intergers?

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Question 414862: The sum of the reciprocals of two consectutive intergers is 3/2.What are the intergers?
Found 2 solutions by MathLover1, Gogonati:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
Let the smaller integer be x, then the larger one is x+%2B+1 (since they are consecutive).
Knowing that the sum of the reciprocals of these two integers is 3 / 2, we obtain an equation:
%281+%2Fx%29+%2B+1%2F%28x+%2B+1%29+=+3+%2F2

%28x+%2B+1+%2B+x%29+%2F+%28x%28x+%2B+1%29%29+=+3%2F+2...cross multiply

2%28x+%2B+1+%2B+x%29+=+3x%28x+%2B+1%29
2x+%2B+2+%2B+2x+=+3x%5E2+%2B+3x
4x+%2B+2++=+3x%5E2+%2B+3x

2++=+3x%5E2+%2B+3x-4x

+3x%5E2+-x+-2+=0.........use quadratic formula
Solved by pluggable solver: Quadratic Formula
Let's use the quadratic formula to solve for x:


Starting with the general quadratic


ax%5E2%2Bbx%2Bc=0


the general solution using the quadratic equation is:


x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29




So lets solve 3%2Ax%5E2%2Bx-2=0 ( notice a=3, b=1, and c=-2)





x+=+%28-1+%2B-+sqrt%28+%281%29%5E2-4%2A3%2A-2+%29%29%2F%282%2A3%29 Plug in a=3, b=1, and c=-2




x+=+%28-1+%2B-+sqrt%28+1-4%2A3%2A-2+%29%29%2F%282%2A3%29 Square 1 to get 1




x+=+%28-1+%2B-+sqrt%28+1%2B24+%29%29%2F%282%2A3%29 Multiply -4%2A-2%2A3 to get 24




x+=+%28-1+%2B-+sqrt%28+25+%29%29%2F%282%2A3%29 Combine like terms in the radicand (everything under the square root)




x+=+%28-1+%2B-+5%29%2F%282%2A3%29 Simplify the square root (note: If you need help with simplifying the square root, check out this solver)




x+=+%28-1+%2B-+5%29%2F6 Multiply 2 and 3 to get 6


So now the expression breaks down into two parts


x+=+%28-1+%2B+5%29%2F6 or x+=+%28-1+-+5%29%2F6


Lets look at the first part:


x=%28-1+%2B+5%29%2F6


x=4%2F6 Add the terms in the numerator

x=2%2F3 Divide


So one answer is

x=2%2F3




Now lets look at the second part:


x=%28-1+-+5%29%2F6


x=-6%2F6 Subtract the terms in the numerator

x=-1 Divide


So another answer is

x=-1


So our solutions are:

x=2%2F3 or x=-1




So our solutions are:
x=2%2F3....is not an integer, so we discard this solution
or x=-1
If the smaller integer is -1, then the second must be -1+%2B+1+=+0
Answer: -1, 0

Answer by Gogonati(855) About Me  (Show Source):
You can put this solution on YOUR website!
1/x+1/x+1=3/2
2(x+1)+2x=3x(x+1)
2x+2+2x=3x^2+3x
3x^2-x-2=0
{-2/3, 1} Answer:The two consecutive integers are 1 and 2
Proof: 1+1/2=3/2