What value of C will provide coincidental roots in the equation $x^3 + 8x^2 + 9x + (18+c)$?

Question

What value of C will provide two coincidental roots in the equation $x^3 + 8x^2 + 9x + (18+c)$?

Answer 1

Do you know about the cubic discriminant? It is a function of the four coefficients, and there is a repeated root if and only if the cubic discriminant is $0$. In this case, solve $$8^29^2-4\cdot9^3-4\cdot8^3(18+c)-27(18+c)^2+18\cdot8\cdot9(18+c)=0$$

There are other approaches where you would not need to know/look up the cubic disciminant.

Answer 2

It means that the equation is of the form: $$(x-a)^2(x-b)=x^3-(2a+b)x^2+(a^2+2ab)x-a^2b = x^3 + 8x^2 + 9x + (18+c)$$

Therefore: $$\begin{cases}2a+b=-8 \\ a^2+2ab=9 \\ a^2 b=-(18+c)\end{cases} $$ Solve for $c$.