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/
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