Question

a, b, c, A and C are real numbers that satisfy the equation

(ax^2+bx+c)^2=Ax^4+16x^3+28x^2+24x+C
How many possible ordered triples of (a,b,c) are there?

Answer 1

well, expand and equate coefficients:

a = A
2ab = 16
2ac+b^2 = 28
2bc = 24
c^2 = C

b=8/A
c=√C

2A√C + 64/A^2 = 28
16/A√C = 24
so
√C = 3A/2

2A(3A/2) + 64/A^2 = 28
A = ±2, ±4/√3
so, find C and thus a,b,c.

Note that c = -√C provides other solutions.