A cubic equation is an equation which can be represented in the form , where are complex numbers and is non-zero. By the fundamental theorem of algebra, cubic equation always has roots, some of which might be equal.
Relation between coefficients and roots:
For a cubic equation , let and be its roots, then the following holds:
|Root expression||Equals to|
This is a special case of Vieta's formulas.
Given that, and are its roots, is the required cubic equation. Since it can be represented in the form , we have the following approach:
Now comparing this with , we have
Find the sum of the squares of the roots of the cubic equation .
We can represent it in the form
Recall that which is an algebraic identity and is not related solely to cubic equations.
From the relations between the coefficients and its roots, we have and . Plugging it in the relation, we have