If 1/x+x=5, what does 1/x^2+x^2 equal?

Date: 08/16/2002 at 23:04:56
From: Shawn Ellis
Subject: Problem on GMAT (algebra)


Dr. Math:

I recently took the GMAT and came across the following problem:

   If 1/x + x = 5, then what does 1/x^2 + x^2 equal?

The answers were 21, 22, 23, 24, 25.

This seemingly easy problem really created a blank for me. I guessed 
25, but I'm still not sure how to work it out.

Any help would be appreciated!

Shawn

Date: 08/17/2002 at 00:47:36
From: Doctor Paul
Subject: Re: Problem on GMAT (algebra)

   1/x = 5-x

so

   1 = 5*x - x^2

   x^2 - 5*x + 1 = 0

solve via quadratic formula

   x = [5 +- sqrt(21)]/2

so

either 

   x = 5/2 + sqrt(21)/2

or

   x = 5/2 - sqrt(21)/2

Now we pick one of these solutions at random and see what happens.

Let's work with the solution x = 5/2 + sqrt(21)/2

Now we compute 1/x^2 + x^2.

First of all, x^2 = 25/4 + 5*sqrt(21)/2 + 21/4 

                  = 46/4 + 5*sqrt(21)/2

                  = [23 + 5*sqrt(21)]/2

Thus 

   1/x^2 + x^2 = 2/[23 + 5*sqrt(21)] + [23 + 5*sqrt(21)]/2

Obtain a common denominator:

   4/(2*[23 + 5*sqrt(21)]) + [23 + 5*sqrt(21)]^2/(2*[23 + 5*sqrt(21)])

= 

   4 + [23 + 5*sqrt(21)]^2
   ----------------------- = 
     2*[23 + 5*sqrt(21)]


   4 + 529 + 230*sqrt(21) + 525 
   ---------------------------- = 
       46 + 10*sqrt(21)

   1058 + 230*sqrt(21)
   ------------------- = 
      46 + 10*sqrt(21)

   23*(46 + 10*sqrt(21)
   -------------------- = 
      (46 + 10*sqrt(21)

   23

If we had chosen instead to work with the solution x = 5/2 - sqrt
(21)/2, the computations would have been different, but the answer 
comes out the same.  Perhaps you can try it on your own.

I hope this helps.  Please write back if you'd like to talk about 
this some more.

- Doctor Paul, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 08/18/2002 at 22:18:58
From: Shawn Ellis
Subject: Thank you (problem on GMAT (algebra))

Doctor Paul:

Thank you so much! You make it look so easy. I now know I need to 
review the quadratic formula, especially when the problems aren't 
nice and tidy.

In your opinion, do you think the average non-math major could do 
this problem in approximately 2 minutes?  (That's what it would take 
to keep pace on the GMAT.)

Thanks again!

Shawn
Tokyo

Date: 08/18/2002 at 23:33:29
From: Doctor Paul
Subject: Re: Thank you (problem on GMAT (algebra))

If your test allows the use of a scientific calculator, then I'd say 
this problem could be done in about 30 seconds.  Once you solve for 
x, you can just plug in 1/x^2 + x^2 and you'll get 23.

Without a calculator, I think it's not reasonable to be done in 2 
minutes. But maybe I did it the hard way. If there's another way 
to do this problem, then 2 minutes might be reasonable.

I hope this helps.  Please write back if you'd like to talk about 
this some more.

- Doctor Paul, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 08/18/2002 at 23:51:46
From: Shawn Ellis
Subject: Problem on GMAT (algebra)

Doctor Paul:

Thank you again for your prompt response!

Calculators aren't allowed when taking the GMAT. So, I would imagine 
there is a simpler method to solve this problem, although I have NO 
idea what that might be...

Shawn
Tokyo

Date: 08/19/2002 at 00:51:15
From: Doctor Greenie
Subject: Re: Problem on GMAT (algebra)

Hi, Shawn -

I enjoyed reading the exchange between you and Dr. Paul on this 
problem.

Yes, there is an easy way to do it, using a "trick" that is a favorite 
of mine.

Given

   1/x + x = 5

simply square both sides:

   (1/x + x)^2 = 5^2

   1/x^2 + 2(1/x)(x) + x^2 = 25

   1/x^2 + 2 + x^2 = 25

   1/x^2 + x^2 = 23

Aggravatingly easy, isn't it...?!

On the other hand, it is always exciting to find a simple way to 
tackle a type of problem which has had you stumped.

- Doctor Greenie, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 08/19/2002 at 01:59:59
From: Shawn Ellis
Subject: Thank you (problem on GMAT (algebra))

Dear Dr. Greenie:

Ah, yes, there it is...  a simpler solution; easily performed within a 
minute!

I know this is exactly what the GMAT Quantitative section is testing 
people on, the ability to see relationships between formulas and find 
the simplest way to solve problems.  

And, this is precisely the ability that I need to improve.

Thanks again.

Shawn
Tokyo