Question

If $P\left(x\right)=a{x}^2+bx+c\text{and}Q\left(x\right)=-a{x}^2+dx+c$, where $ac\ne 0$ [a, b, c, d are all real], then P(x).Q(x) = 0 has

• A. at least two real roots
• B. two real roots
• C. four real roots
• D. no real root

Answer 1

The correct option is A at least two real roots

$P\left(x\right)=a{x}^2+bx+cQ\left(x\right)=-a{x}^2+dx+c{D}_1=b^2-4ac{D}_2=d^2+4ac$

If $ac$ is negative, then $D_1$ will have 2 real roots ($D_2$ too can have real roots)

If $ac$ is positive,then $D_2$ will have 2 real roots ($D_1$ too can have real roots)