SOLUTION: Two positive numbers differ by 4. If the sum of their reciprocals is 2/3, find these two numbers.

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Question 860417: Two positive numbers differ by 4. If the sum of their reciprocals is 2/3, find these two numbers.
Answer by ben720(159) About Me  (Show Source):
You can put this solution on YOUR website!
If the two positive numbers differ by four,
Let x = 1st number and x+4 = 2nd.
The sum of their reciprocals, 1%2Fx%2B1%2F%28x%2B4%29 equals 2%2F3
1%2Fx%2B1%2F%28x%2B4%29=2%2F3
To isolate x, first multiply both sides by x(x+4)
x%2B4%2Bx=%282x%29%28x%2B4%29%2F3
Simplify
2x%2B4+=+%282x%5E2%2B8x%29%2F3
Multiply both sides by 3
6x%2B12+=+2x%5E2%2B8x
Subtract (6x+12) from both sides
0=2x%5E2%2B2x-12
Use the quadratic formula

Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 2x%5E2%2B2x%2B-12+=+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=%282%29%5E2-4%2A2%2A-12=100.

Discriminant d=100 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-2%2B-sqrt%28+100+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%282%29%2Bsqrt%28+100+%29%29%2F2%5C2+=+2
x%5B2%5D+=+%28-%282%29-sqrt%28+100+%29%29%2F2%5C2+=+-3

Quadratic expression 2x%5E2%2B2x%2B-12 can be factored:
2x%5E2%2B2x%2B-12+=+2%28x-2%29%2A%28x--3%29
Again, the answer is: 2, -3. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+2%2Ax%5E2%2B2%2Ax%2B-12+%29


As shown by this solver, the answers for x are either 2 or -3.
However, as both numbers must be positive, it must be 2.
Your answers are 2 and 6 (as 2+4=6)