SOLUTION: The sum of a number and its reciprocal is 85/42. Find the number.

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Question 541670: The sum of a number and its reciprocal is 85/42. Find the number.
Answer by neatmath(302) About Me  (Show Source):
You can put this solution on YOUR website!

Let's first set up an equation using x as our variable:

x%2B1%2Fx=85%2F42

Then we just need to solve for x:

42x%2B42%2Fx=85

42x%5E2%2B42x%2Fx=85x

42x%5E2%2B42=85x

42x%5E2-85x%2B42=0

Then we can use the quadratic formula to solve for x:

Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 42x%5E2%2B-85x%2B42+=+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=%28-85%29%5E2-4%2A42%2A42=169.

Discriminant d=169 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--85%2B-sqrt%28+169+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-85%29%2Bsqrt%28+169+%29%29%2F2%5C42+=+1.16666666666667
x%5B2%5D+=+%28-%28-85%29-sqrt%28+169+%29%29%2F2%5C42+=+0.857142857142857

Quadratic expression 42x%5E2%2B-85x%2B42 can be factored:
42x%5E2%2B-85x%2B42+=+42%28x-1.16666666666667%29%2A%28x-0.857142857142857%29
Again, the answer is: 1.16666666666667, 0.857142857142857. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+42%2Ax%5E2%2B-85%2Ax%2B42+%29


This gives us a good approximation to work with.

Using an online factoring calculator, we find that our equation factors as:

%287x-6%29%286x-7%29=0

From this we can easily see that x=6%2F7 or x=7%2F6

It would take us a BIT of time to factor that equation properly and get the exact answer.

I'm trying to think of other ways off the top of my head to solve this.

I hope this helps!