Question

Let $f ( x )= x ^2+ ax + b$ and $g ( x )= x ^2+ cx + d$ be two quadratic polynomials with real coefficients and satisfy $a c=2(b+d)$. 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 $a^2<4 b$
and $ c ^2<4 d$
(adding), ______
$\therefore a ^2+ c ^2<4( b + d ) \Rightarrow a ^2+ c ^2<2 ac \Rightarrow( a - c )^2<0$, 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.