You are here:

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

Questioner:Isaac
Country:Illinois, United States
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.

Volunteer

#### Paul Klarreich

##### Expertise

I can answer questions in basic to advanced algebra (theory of equations, complex numbers), precalculus (functions, graphs, exponential, logarithmic, and trigonometric functions and identities), basic probability, and finite mathematics, including mathematical induction. I can also try (but not guarantee) to answer questions on Abstract Algebra -- groups, rings, etc. and Analysis -- sequences, limits, continuity. I won't understand specialized engineering or business jargon.

##### Experience

I taught at a two-year college for 25 years, including all subjects from algebra to third-semester calculus.

Education/Credentials
-----------