SOLUTION: The reciprocal of an integer plus the reciprocal of two times the integer plus two equals 2/3. Find the integer

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Question 159837: The reciprocal of an integer plus the reciprocal of two times the integer plus two equals 2/3. Find the integer
Answer by KnightOwlTutor(293) About Me  (Show Source):
You can put this solution on YOUR website!
X=unknown integer
1/X+1/2X+2=2/3

The common denominator for all is 3(X)(2X+2)
Multiply each term by the common denominator
3(2X+2)+3(X)=2(X)(2X+2)
use the distributive property
6X+6+3X=4x^2+4X
9X+6=4X^2+4X
subtract 9X from each side
6=4X^2-5X
Subtract 6 from both sides
0=4X^2-5X-6
This is a quadratic equation
Use the quadratic formula to solve
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 4x%5E2%2B-5x%2B-6+=+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=%28-5%29%5E2-4%2A4%2A-6=121.

Discriminant d=121 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--5%2B-sqrt%28+121+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-5%29%2Bsqrt%28+121+%29%29%2F2%5C4+=+2
x%5B2%5D+=+%28-%28-5%29-sqrt%28+121+%29%29%2F2%5C4+=+-0.75

Quadratic expression 4x%5E2%2B-5x%2B-6 can be factored:
4x%5E2%2B-5x%2B-6+=+%28x-2%29%2A%28x--0.75%29
Again, the answer is: 2, -0.75. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+4%2Ax%5E2%2B-5%2Ax%2B-6+%29


X=2
Plug in 2 for the variable x in the original equatio
1/2+(1/4+2)=2/3
1/2+1/6+2/3
the common denominator is 6
3/6+1/6=4/6 simplify the term to 2/3 by dividing both numerator and denominator by 2