SOLUTION: the difference between two whole numbers is 8 and the difference between their reciprocals is 1/6. find the two numbers

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Question 221043: the difference between two whole numbers is 8 and the difference between their reciprocals is 1/6. find the two numbers
Found 2 solutions by drj, jim_thompson5910:
Answer by drj(1380) About Me  (Show Source):
You can put this solution on YOUR website!
The difference between two whole numbers is 8 and the difference between their reciprocals is 1/6. find the two numbers.

Step 1. Let x be one whole number and 1%2Fx be its reciprocal.

Step 2. Let 8-x be the other whole number and 1%2F%288-x%29 be its reciprocal.

Step 3. Then, 1%2F%288-x%29-1%2Fx=1%2F6

Step 4. Multiply by 6x(8-x) to both sides of the equation to get rid of denominators.

6x%288-x%29%2A1%2F%288-x%29-6x%288-x%29%2A1%2Fx=6x%288-x%29%2A1%2F6

6x-6%288-x%29=x%288-x%29

12x-48=8x-x%5E2

Add x%5E2-8x to both sides of the equation to get a quadratic equation

12x-48%2Bx%5E2-8x=8x-x%5E2%2Bx%5E2-8x

x%5E2%2B4x-48=0

Step 5. To solve the equation, use the quadratic formula x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+ where

a=1, b=4 and c=-48.

Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B4x%2B-48+=+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=%284%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.

x%5B1%5D+=+%28-%284%29%2Bsqrt%28+208+%29%29%2F2%5C1+=+5.21110255092798
x%5B2%5D+=+%28-%284%29-sqrt%28+208+%29%29%2F2%5C1+=+-9.21110255092798

Quadratic expression 1x%5E2%2B4x%2B-48 can be factored:
1x%5E2%2B4x%2B-48+=+1%28x-5.21110255092798%29%2A%28x--9.21110255092798%29
Again, the answer is: 5.21110255092798, -9.21110255092798. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B4%2Ax%2B-48+%29



I am unable to get whole numbers based on your problem statement, please check the problem or resubmit.

I hope the above steps were helpful.

For FREE Step-By-Step videos in Introduction to Algebra, please visit http://www.FreedomUniversity.TV/courses/IntroAlgebra and for Trigonometry visit http://www.FreedomUniversity.TV/courses/Trigonometry.

Good luck in your studies!

Respectfully,
Dr J







Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Let's assume that these numbers are positive.

"the difference between two whole numbers is 8 " means that x-y=8. Note: I'm implying that x%3Ey. Solve this equation for 'y' to get y=x-8

Also, " the difference between their reciprocals is 1/6" means that 1%2Fy-1%2Fx=1%2F6 (note: since x%3Ey, this means 1%2Fx%3C1%2Fy)


1%2Fy-1%2Fx=1%2F6 Start with the second equation.


1%2F%28x-8%29-1%2Fx=1%2F6 Plug in y=x-8


Multiply EVERY term by the LCD 6x%28x-8%29 to clear out the fractions.

6x%281%29-6%28x-8%29=x%28x-8%29 Simplify


6x-6%28x-8%29=x%28x-8%29 Multiply


6x-6x%2B48=x%5E2-8x Distribute


6x-6x%2B48-x%5E2%2B8x=0 Get everything to one side.


-x%5E2%2B8x%2B48=0 Combine like terms.


Notice that the quadratic -x%5E2%2B8x%2B48 is in the form of Ax%5E2%2BBx%2BC where A=-1, B=8, and C=48


Let's use the quadratic formula to solve for "x":


x+=+%28-B+%2B-+sqrt%28+B%5E2-4AC+%29%29%2F%282A%29 Start with the quadratic formula


x+=+%28-%288%29+%2B-+sqrt%28+%288%29%5E2-4%28-1%29%2848%29+%29%29%2F%282%28-1%29%29 Plug in A=-1, B=8, and C=48


x+=+%28-8+%2B-+sqrt%28+64-4%28-1%29%2848%29+%29%29%2F%282%28-1%29%29 Square 8 to get 64.


x+=+%28-8+%2B-+sqrt%28+64--192+%29%29%2F%282%28-1%29%29 Multiply 4%28-1%29%2848%29 to get -192


x+=+%28-8+%2B-+sqrt%28+64%2B192+%29%29%2F%282%28-1%29%29 Rewrite sqrt%2864--192%29 as sqrt%2864%2B192%29


x+=+%28-8+%2B-+sqrt%28+256+%29%29%2F%282%28-1%29%29 Add 64 to 192 to get 256


x+=+%28-8+%2B-+sqrt%28+256+%29%29%2F%28-2%29 Multiply 2 and -1 to get -2.


x+=+%28-8+%2B-+16%29%2F%28-2%29 Take the square root of 256 to get 16.


x+=+%28-8+%2B+16%29%2F%28-2%29 or x+=+%28-8+-+16%29%2F%28-2%29 Break up the expression.


x+=+%288%29%2F%28-2%29 or x+=++%28-24%29%2F%28-2%29 Combine like terms.


x+=+-4 or x+=+12 Simplify.


So the solutions are x+=+-4 or x+=+12

But let's ignore the negative solution.


So this means that the two numbers are 12 and 4.