SOLUTION: The sum of a number and it's reciprocal is 10 over 3. What is the number?

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Question 307020: The sum of a number and it's reciprocal is 10 over 3. What is the number?
Found 2 solutions by themathtutor2009, rapaljer:
Answer by themathtutor2009(81) About Me  (Show Source):
You can put this solution on YOUR website!
x+1/x=10/3

(x^2+1)/x=10/3

3(x^2+1)=10x

3x^2+3=10x

3x^2-10x+3=0

(x-3)(3x-1)=0

x-3=0 or 3x-1=0

x=3 or x=1/3

Number is 3 or 1/3

Answer by rapaljer(4671) About Me  (Show Source):
You can put this solution on YOUR website!
Let x = the number
1%2Fx+= the reciprocal of the number

The equation is
x+%2B+1%2Fx+=+10%2F3

The LCD = 3x, so multiply both sides by 3x:
3x%2Ax+%2B+3x%2A%281%2Fx%29+=+3x%2A%2810%2F3%29
3x%5E2+%2B+3+=+10x

This is a quadratic equation, so set it equal to zero:
3x%5E2+-10x+%2B+3=0

It just happens to factor!! (Isn't that nice??)
%283x-1%29%28x-3%29=0

(By the way, if ANYONE needs help factoring this, send me an Email at rapaljer@seminolestate.edu, and I'll be glad to help you find the explanation on my website!!!)

There are 2 solutions to the quadratic equation:
FIRST SOLUTION:
3x-1=0
3x=1
x=1%2F3

SECOND SOLUTION:
x-3=0
x=3

There really are NOT two solutions to this problem, since the second solution is actually the RECIPROCAL of the first solution, and vice-versa.

For additional help with this topic, I have several sections on my website that may be useful. You can find my website by doing a Google search for my name "Dr. Robert J. Rapalje". Look for "Rapalje Homepage", which should be the first item on the search. Once you find my homepage, for a written explanation, click on "Basic, Intermediate, and College Algebra: One Step at a Time." For "Factoring Trinomials" look in Basic Algebra in Chapter 2 for the sections on Trinomial Factoring and Advanced Trinomial Factoring. For help with the topic of solving Quadratic Equations, this is also in Basic Algebra, Chapter 2. There you will find a detailed explanation, and solutions to problems in LIVING COLOR!!

To see my FREE video explanation on Factoring, look for the link on my homepage called "Rapalje Videos in Living Color." Again, choose either Basic Algebra or Intermediate Algebra, and look for the topic of "Factoring."

I have many other topics explained in this way from Basic, Intermediate, and College Algebra. Please check out my website to see what I have. It's all FREE, and if you send me an Email, I'll be glad to help you find something. I'm not selling anything. If you get a good explanation, math is usually not as hard as you think it is.

Dr. Robert J. Rapalje, Retired
Seminole State College of Florida
Altamonte Springs Campus