Polynomial Having Common Root

Question

If the equation $ax^3+3bx^2+3cx+d=0$ and $ax^2 +2bx+c=0$ have a common root then what is $\dfrac{(bc-ad)^2}{(ac-b^2)(bd-c^2)}$?

Answer 1

Multiply the second quadratic equation by $x$ and subtract it from the first cubic equation. This will land you with another quadratic equation,
$bx^2 + 2cx + d = 0$

This equation and the original quadratic equation ($ax^2 + 2bx + c = 0$) too share a common root. Hence applying the condition of a common root on the 2 quadratic equations,

$(2c^2 - 2bd)(2b^2 - 2ac) = (da-bc)^2$.

Simplify this equation to get 4 as the answer to your question.

Also see this: Condition for a common root in two given quadratic equations