Quadratic Problem: Find Sum of Roots of 4 Equations

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Homework Help Overview

The discussion revolves around a problem involving quadratic equations where the roots of two equations are interrelated. The equations are structured such that the roots of the first equation depend on the coefficients influenced by the roots of the second equation, leading to a complex relationship among four distinct variables: a, b, c, and d.

Discussion Character

  • Exploratory, Mathematical reasoning, Problem interpretation

Approaches and Questions Raised

  • Participants explore the relationships between the roots and coefficients of the quadratic equations, noting that traditional elimination methods may not be effective. There are attempts to isolate variables and substitute them into other equations to simplify the problem.

Discussion Status

Some participants have provided insights into the relationships between the variables, suggesting that the sum of certain pairs of roots can lead to a simplified expression. There is ongoing exploration of how to manipulate the equations to derive further insights, but no consensus on a final solution has been reached.

Contextual Notes

Participants note that the variables a, b, c, and d are distinct, which may impose additional constraints on the problem-solving approach. The complexity of the relationships among the equations is acknowledged, indicating that straightforward methods may not apply.

erisedk
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Homework Statement



If roots of the equation ##x^2 - 10cx - 11d = 0## are ##a, b## and those of ## x^2 - 10ax - 11 b = 0## are ##c, d## then the value of ##a + b + c + d## is (##a, b, c## and ##d## are distinct numbers)

Homework Equations

The Attempt at a Solution


##a+b=10c##
##c+d=10a##
## ab=-11d##
##cd=-11b##

Four equations, four unknowns. Obviously, this isn't supposed to be solved using regular elimination. It gets way too terrible. I can't think of a better way though. Please help.
 
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10(a + c)
 
erisedk said:

Homework Statement



If roots of the equation ##x^2 - 10cx - 11d = 0## are ##a, b## and those of ## x^2 - 10ax - 11 b = 0## are ##c, d## then the value of ##a + b + c + d## is (##a, b, c## and ##d## are distinct numbers)

Homework Equations

The Attempt at a Solution


##a+b=10c##
##c+d=10a##
## ab=-11d##
##cd=-11b##

Four equations, four unknowns. Obviously, this isn't supposed to be solved using regular elimination. It gets way too terrible. I can't think of a better way though. Please help.
@Math_QED is right, the sum is 10(a+c). So isolate b and d from the first two equations and substitute into the third and fourth. See what you get for a+c.
 
Thank you!
b = 10c - a
d = 10a - c
a(10c - a) = -11d
c(10a - c) = -11b
c2 - a2 = 11(b - d)
a + c = 121
b + d = 9 (a + c)
So a + b + c + d = 1210
 
erisedk said:
Thank you!
b = 10c - a
d = 10a - c
a(10c - a) = -11d
c(10a - c) = -11b
c2 - a2 = 11(b - d)
You can divide the equation with c-a as a,b,c,d are all different numbers.
erisedk said:
a + c = 121
b + d = 9 (a + c)
So a + b + c + d = 1210

Well done!
 

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