For what rational number c do the equations...?

Question

For what rational number c do the equations

x^3 + cx^2 + 3 = 0 and x^2 + cx + 1 = 0 have a common solution?

Answer 1

write your equations as:

cx^2=-3-x^3  and cx=-1-x^2

multiply the equation on the right by x to get:

cx^2=-x-x^3

and notice that we can equate cx^2 to get:

-3-x^3=-x-x^3 which means that x=3 is a common solution

we can go to the second equation above and substitute x=3 to get:

3^2+3c+1=0 or c=-10/3

for this value of c you will get the common root x=3

Answer 2

cx^2 = -x^3 - 3 (eq.1)

cx = -x^2 - 1 (eq.2)

multiply eq.2 by x:

-x^3 - x = cx^2

-x^3 - 3 = cx^2

subtract the two equations:

-x+3 = 0

-x = -3

x = 3

9 + 3c + 1 = 0

3c + 10 = 0

3c = -10

c = -10/3