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### Equations with a Common Root

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

Dr. Math,

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/
```
Associated Topics:
High School Number Theory

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