Question

If $P\left(x\right)=a{x}^2+bx+c$ and $Q\left(x\right)=-a{x}^2+dx+c$ , $ac\ne 0$, then the equation $P\left(x\right).Q\left(x\right)=0$ has

• A. two imaginary roots
• B. more than two imaginary roots
• C. atleast two real roots
• D. no real roots

Answer 1

The correct option is C atleast two real roots

$P\left(x\right)=a{x}^2+bx+c$
Hence, for real roots,
$b^2-4ac\ge 0$
$Q\left(x\right)=-a{x}^2+dx+c$
Hence, for real roots,
$d^2+4ac\ge 0$
Now,
$ac\ne 0$
Now if $ac<0$
Then,
$b^2-4ac>0$
Hence, $P\left(x\right).Q\left(x\right)$ has atleast $2$ real roots .
If $ac>0$
Then
$d^2+4ac>0$
Hence, $P\left(x\right).Q\left(x\right)$ has atleast $2$ real roots .