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

#1**+2 **

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

CPhill
Jan 6, 2018