# Two Cubic Equations

##### Age 16 to 18 ShortChallenge Level

The positive real numbers $a$, $b$ and $c$ are such that the equation $$x^3 +ax^2=bx+c$$ has three real roots, one positive and two negative.

Which one of the following correctly describes the real roots of the equation $$x^3+c=ax^2+bx?$$

(A) It has three real roots, one positive and two negative.

(B) It has three real roots, two positive and one negative.

(C) It has three real roots, but their signs differ depending on ܽ$a$, $b$, and ܿ$c$.

(D) It has exactly one real root, which is positive.

(E) It has exactly one real root, which is negative.

(F) It has exactly one real root, whose sign differs depending on ܽ$a$, $b$, and ܿ$c$.

(G) The number of real roots can be one or three, but the number of roots differs depending on ܽ$a$, $b$, and ܿ$c$.

There are some hints in the Getting Started section.

This question comes from TMUA Specimen paper 2.

The Test of Mathematics for University Admission (or TMUA) is designed to give you the opportunity to demonstrate that you have the essential mathematical thinking and reasoning skills needed for a demanding undergraduate Mathematics or Mathematics-related course. There are several UK universities which encourage their applicants to sit TMUA, and a good performance may result in a reduced offer.