+884 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.
+128408 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
