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Eric and Charles each think of a quadratic polynomial. To their surprise, both quadratics start  \(x^2+4x+\cdots\). The ratio of the discriminant , \(b^2-4ac\) , of Eric's polynomial to the discriminant of Charles's polynomial is equal to the ratio of Charles's constant term to Eric's constant term. If their constant terms are not equal, find the sum of the constant terms.

ant101  Jan 6, 2018
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Let C1  be  the constant in Eric's polynomial  and C2  be the constant in Charles's polynomial

 

And we have that

 

[16 - 4C1] / [16 - 4C2]  =  C2 /C1     cross-multiply

 

16C1   -  4C1^2   =  16C2  - 4C2^2

 

4C1 ( 4 - C1) =  4C2 ( 4 - C2)       divide by 4

 

C1 (4 - C1) =  C2 (4 - C2)      rearrange   as

 

C2 / C1  =  (4 - C1) / (4 - C2)  implies that

 

C2 =  4 - C1       and    C1  = 4 - C2      which  both imply that

 

C1 + C2  =   4

 

 

cool cool cool

CPhill  Jan 6, 2018

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