SOLUTION: Find two positive numbers that differ by eight and whose reciprocals sum is 1/6

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Question 944234: Find two positive numbers that differ by eight and whose reciprocals sum is 1/6
Found 2 solutions by macston, MathLover1:
Answer by macston(5194) About Me  (Show Source):
You can put this solution on YOUR website!
X=one positive number, Y is the other positive number
X-Y=8; 1%2FX+%2B+1%2FY+=1%2F6
X=Y+8 Add Y to each side of Equation 1 and substitute in Equation 2
1%2F%28Y%2B8%29%2B1%2FY=1%2F6 Multiply each side by 6
6%2F%28Y%2B8%29%2B6%2FY=1 Subtract 6/Y from each side
6%2F%28Y%2B8%29=1-6%2FY convert right side to fraction
6%2F%28Y%2B8%29=1%28Y%2FY%29-6%2FY
6%2F%28Y%2B8%29=Y%2FY-6%2FY
6%2F%28Y%2B8%29=%28Y-6%29%2FY Multiply each side by Y+8
6=%28Y-6%29%28Y%2B8%29%2FY Multiply each side by Y
6Y=%28Y-6%29%28Y%2B8%29 Multiply right side through
6Y=Y%5E2%2B2Y-48 Subtract 6Y from each side
0=y%5E2-4Y-48
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation aY%5E2%2BbY%2Bc=0 (in our case 1Y%5E2%2B-4Y%2B-48+=+0) has the following solutons:

Y%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=%28-4%29%5E2-4%2A1%2A-48=208.

Discriminant d=208 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--4%2B-sqrt%28+208+%29%29%2F2%5Ca.

Y%5B1%5D+=+%28-%28-4%29%2Bsqrt%28+208+%29%29%2F2%5C1+=+9.21110255092798
Y%5B2%5D+=+%28-%28-4%29-sqrt%28+208+%29%29%2F2%5C1+=+-5.21110255092798

Quadratic expression 1Y%5E2%2B-4Y%2B-48 can be factored:
1Y%5E2%2B-4Y%2B-48+=+1%28Y-9.21110255092798%29%2A%28Y--5.21110255092798%29
Again, the answer is: 9.21110255092798, -5.21110255092798. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-4%2Ax%2B-48+%29

The positive answer is 9.2111 so ANSWER 1 Y=9.2111
X-Y=8
X-9.2111=8 Add 9.2111 to each side
X=17.2111 ANSWER 2 X=17.2111
FINAL ANSWER THE TWO POSITIVE NUMBERS ARE 17.2111 and 9.2111
CHECK
1/X+1/Y=1/6
1/17.2111+1/9.2111=1/6
0.16667=1/6
0.16667=0.16667






Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!

let two positive numbers be x and y
if they differ by eight, we have
x-y=8 => x=y%2B8......eq.1
and if their reciprocals sum is 1%2F6, we have

1%2Fx%2B1%2Fy=1%2F6.
y%2Fxy%2Bx%2Fxy=1%2F6
%28y%2Bx%29%2Fxy=1%2F6.....cross multiply
6%28y%2Bx%29=1%2Axy
6y%2B6x=xy.....eq.2 ....substitute x from eq.1
6y%2B6%28y%2B8%29=%28y%2B8%29y
6y%2B6y%2B48=y%5E2%2B8y
12y%2B48=y%5E2%2B8y
0=y%5E2%2B8y-12y-48
y%5E2-4y-48=0.....use quadratic formula
y+=+2%2B2sqrt%2813%29=9.211102550927979
y = 2-2 sqrt(13)=-5.2 we need positive number only
find x:
x=y%2B8=> x=9.2%2B8=17.211102550927979

check the sum of reciprocals:
1%2Fx%2B1%2Fy=1%2F6
1%2F17.211102550927979%2B1%2F9.211102550927979=0.1666666666666667
0.0581020301890005%2B0.1085646364776662=0.1666666666666667
0.1666666666666667=0.1666666666666667