Question

y=x^3=cx and y=ax^2+bx-1 have a common tangent at (-1,-4). find a, b, andc

Answer 1

I will assume the first equation is
y = x^3 + cx (since the + and + are just a "shift" away)

since (-1,-4) is a common point
-4 = -1 - c
c = 3 , and

-4 = a(1) -b - 1
a-b = -3 (#1)

If they have a common tangent, then their slopes must be the same.
3x^2 + c = 2ax + b , but c = 3
3x^2 + 3 = 2ax + b
again (-1,-4) must satisfy.
3 + 3 = -2a + b
2a - b = -6 (#2)

#2 - #1 ---> a = -3
back in #1
-3 - b = -3
b = 0

a= -3, b=0, and c=3