Equations with a Common Root

Date: 08/22/2001 at 05:06:20
From: William Sandford
Subject: Simultaneous equations

Dr. Math,

Can you please help me figure out the following question? It's too 
hard for me. Thanks!

Find all real numbers a such that the equations

x^9+ax^7-(a-3)x^6-1/2x^2+1=0
2x^5+2ax^3-(2a-6)x^2+1=0

have a common root.

Help would be greatly appreciated.
Regards,
William

Date: 08/22/2001 at 16:54:22
From: Doctor Jaffee
Subject: Re: Simultaneous equations

Hi William,

I noticed that the exponents of x in the first three terms of the 
first equation are 9, 7, and 6.

The corresponding exponents in the second equation are 5, 3, and 2.

In both cases the differences between the corresponding adjacent 
exponents are the same. That gave me the idea of multiplying each term 
in the second equation by x^4.

Then I noticed that the coefficients of the first three terms in the 
second equation were twice as large as the corresponding coefficients 
in the first equation, so I multiplied each term in the first equation 
by 2.

Then, when I subtracted the two equations, I got a fourth-degree 
equation in quadratic form that I could solve for x.  

Once I knew the value for x, I could find the value for a that would 
work.

So, if all of this makes sense to you, give it a try and see if you 
can find the solution. If you want to check your solution with me, 
write back. If you have difficulties, let me know and show me what you 
have done so far and I'll try to help.

This is a very challenging problem for a thirteen-year-old, but if you 
understand my hints, you can solve it.

Good luck. 

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