Q) If f(x) = ax^2 +bx+c AND g(x) = -ax^2+bx+c where ac is not equal to 0, then what can be said about the roots of the equation f(x) .g(x) = 0 ?

A)
f(x).g(x) = -a^2.x^4 + (bx + c)^2

f(x).g(x) = a^2.x^4 - b^2.x^2 - 2bc.x - c^2 = 0

Using Descartes' Rule of signs.... there is one positive root, and a max of 3 negative roots (can be 1 negative root and rest 2 imaginary). Remember that imaginary roots occur in pairs always.
So the least number of real roots is 1 positive and 1 negative... so at least 2 real roots.

So the answer is at least 2 real roots.

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