If $P(x)=ax^2+bx+c$ and $Q(x)=-ax^2+dx+c$ , $ac\neq 0$, then the equation $P(x). Q(x)=0$ has
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two imaginary roots
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more than two imaginary roots
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atleast two real roots
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no real roots
P(x)Q(x) = -(ax^2+bx+c)(ax^2-dx-c) = 0. This is a product of two quadratics. Since the leading coefficients of the two quadratics have opposite signs (a and -a), the product will have at least two real roots because the function will cross the x-axis.
Since ac is not equal to 0, both P(x) and Q(x) are valid quadratic equations. The discriminant of P(x) is b^2 - 4ac, and the discriminant of Q(x) is d^2 + 4ac. If P(x) does not have real roots, its discriminant is negative, which implies 4ac is positive, making the discriminant of Q(x) strictly positive. Therefore, at least one of these two equations must have real roots, ensuring the combined equation P(x)Q(x) = 0 has at least two real roots.