Question

Let $f(x)=x^2+ax+b$ and $g(x)=x^2+cx+d$ be two quadratic polynomials with real coefficients and satisfy $ac=2(b+d)$. Then which of the following is(are) correct?

• A. Exactly one of either $f(x)=0$ or $g(x)=0$ must have real roots.
• B. Atleast one of either $f(x)=0$ or $g(x)=0$ must have real roots.
• C. Both $f(x)=0$ and $g(x)=0$ must have real roots.
• D. Both $f(x)=0$ and $g(x)=0$ must have imaginary roots.

Answer 1

Answer: (B)

Let $f(x)=0$ and $g(x)=0$ have both imaginary roots. Hence $a^2<4b$
and $c^2<4d$
(adding), ______
$\therefore a^2+c^2<4(b+d)\Rightarrow a^2+c^2<2ac\Rightarrow (a-c{)}^2<0$, which is not possible.
Hence both $f(x)=0$ and $g(x)=0$ can not have imaginary roots $\Rightarrow$ At least one of the equations must have real roots.