What are the conditions that the following equation has four real roots? (Tick all that apply)
$$ (a_1x^2+b_1x+c_1)(a_2x^2+b_2x+c_2) = 0$$
a. $c_1c_2 > 0$b. $a_1c_2 < 0$
c. $a_2c_1 < 0$
My Attempt: For the equation to have four real roots, both the equations should have two real roots each. From their discriminants, this gives us
$$b_1^2 – 4a_1c_1 > 0 \text{ and } b_2^2 – 4a_2c_2 > 0$$
I could not move past this step or use this step to derive any of the expressions mentioned.
Best Answer
It is sufficient if $a_1c_1 < 0$ and $a_2c_2< 0$.
If all 3 conditions hold then, multiplying the first and second $a_1c_1c_2^2 < 0$ so $a_1c_1 < 0$.
Similarly, multiplying the first and third, $a_2c_2c_1^2 < 0$ so $a_2c_2< 0$.
Therefore if all three conditions hold there are four real roots. The roots may not all be different depending on the coefficients.