Question

If $f(x) = ax^2 + bx + c$ and $g(x) = -ax^2 + bx + c$ where $ac \neq 0$ then $f(x), g(x) = 0$ has:

• A. Two real roots
• B. four real roots
• C. atleast two real roots
• D. atmost two real roots

Answer 1

Answer: (C)

atleast two real roots

Answer 2

The discriminants are $\Delta_f = b^2 - 4ac$ and $\Delta_g = b^2 + 4ac$. Since $ac \neq 0$, either $ac < 0$ or $ac > 0$. If $ac < 0$, then $-4ac > 0$, so $\Delta_f = b^2 - 4ac > 0$. Thus $f(x)=0$ has two distinct real roots. If $ac > 0$, then $4ac > 0$, so $\Delta_g = b^2 + 4ac > 0$. Thus $g(x)=0$ has two distinct real roots. In either case, at least one of the equations has two distinct real roots.