SOLUTION: If the sum of a rational number and its reciprocal is 13/6,then the number is?

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Question 982120: If the sum of a rational number and its reciprocal is 13/6,then the number is?
Found 2 solutions by ikleyn, macston:
Answer by ikleyn(52855) About Me  (Show Source):
You can put this solution on YOUR website!

Let  x  be our rational number.
Then its reciprocal is  1%2Fx,  and the equation is
x + 1%2Fx = 13%2F6.

Multiply both sides by  6x.  You will get a quadratic equation

6x%5E2 + 6 = 13x,  or

6x%5E2 - 13x + 6 = 0.

Solve it using the quadratic formula.  You will get the roots

x%5B1%5D = 3%2F2  and  x%5B2%5D = 2%2F3.

Check:   3%2F2 + 2%2F3 = %289%2B4%29%2F6 = 13%2F6.

Answer.  x%5B1%5D = 3%2F2 and x%5B2%5D = 2%2F3.


Answer by macston(5194) About Me  (Show Source):
You can put this solution on YOUR website!
.
x=number
.
x%2B1%2Fx=13%2F6
x%5E2%2B1=%2813%2F6%29x
x%5E2-%2813%2F6%29x%2B1=0
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B-2.16666666666667x%2B1+=+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=%28-2.16666666666667%29%5E2-4%2A1%2A1=0.694444444444459.

Discriminant d=0.694444444444459 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--2.16666666666667%2B-sqrt%28+0.694444444444459+%29%29%2F2%5Ca.




Quadratic expression 1x%5E2%2B-2.16666666666667x%2B1 can be factored:
1x%5E2%2B-2.16666666666667x%2B1+=+1%28x-1.50000000000001%29%2A%28x-0.666666666666664%29
Again, the answer is: 1.50000000000001, 0.666666666666664. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-2.16666666666667%2Ax%2B1+%29

ANSWER: x=1.5 or x=2/3