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I'm part of a math club and this is one of the problems that we were asked to solve. I'm having a little bit of trouble with it.

Let a,b,c,d represent 4 different non-zero integers such that the absolute value of each integer is less than 11. If c and d are the solutions for x of x^2+ax+b=0 and if a and b are the solutions for x of 2x^2-cx-20d=0, find the value of a+b+c+d.

Thanks a lot.

Sincerely,
Isaac

Answer
Questioner:Isaac
Country:Illinois, United States
Category:Advanced Math
Private:No <<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<< changed
Subject:Difficult polynomial problem
Question:

I'm part of a math club and this is one of the problems that we were asked to solve. I'm having a little bit of trouble with it.

Let a,b,c,d represent 4 different non-zero integers such that the absolute value of each integer is less than 11. If c and d are the solutions for x of x^2+ax+b=0 and if a and b are the solutions for x of 2x^2-cx-20d=0, find the value of a+b+c+d.

Thanks a lot.

Sincerely,
Isaac

..........................................
I don't have a quick and elegant solution for you, but I will assume that you know:
A. Formulas for sum and product of the roots of a quadratic.
B. Rational root theorem for polynomial equations.

If c and d are the solutions for x of x^2+ax+b=0

1.sum of roots:  c + d = - a
2.product :      cd = b

If a and b are the solutions for x of 2x^2-cx-20d=0

3.sum of roots:  a + b = c/2
4.product:       ab = -10d

"Add" 1,-2:      c + d - cd = -a - b = - (a + b) = - c/2
Multiply 1,2: (c + d)(cd) = - ab = +10d


You have two equations in c,d:

cd - c - d = c/2   (A)
(c + d)cd = 10d    (B)
(c + d)c = 10    (also B)

Solve A for d:
2cd - 2c - 2d = c

2cd - 3c - 2d = 0
2cd - 2d = 3c

2d(c - 1) = 3c
      3c
d = --------
   2c - 2

Subst into B:

(c + d)c = 10

       3c
(c + -------)c = 10
    2c - 2


2c^2 - 2c + 3c
(--------------)c = 10
    2c - 2


(2c^2 + c)c = 10(2c - 2)
    
2c^3 + c^2 = 20c - 20

2c^3 + c^2 - 20c + 20

Possible roots are   1,2,4,5,10

Try c = 2  ----- it works!

16 + 4 - 40 + 20 = 0

      6
d = -------
   4 - 2

d = 3


3.sum of roots:  a + b = 1
4.product:       ab = -30

Looks like a,b = -5, +6.

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