Quick questions on Sums and products of zeroes: relating coefficients to the elementary symmetric functions of the roots for quadratics, cubics and quartics, and using them to find missing zeroes, evaluate symmetric expressions, find unknown coefficients, and handle roots in arithmetic or geometric progression

The real power is that any expression in the roots that is symmetric (unchanged if you swap the roots around) can be rewritten in terms of the sum, the product and the pair-sum, then evaluated by substitution. You never find the roots. The two identities that cover almost every Year 11 question are:

The same naming idea handles a stated relationship between roots. If you are told one root is the negative of another, name the roots $a$, $-a$, $b$: the opposite pair cancels in the sum, so the sum gives $b$, and the pair-sum reduces to $-a^2$, giving $a$. If you are told two roots are equal, name them $a$, $a$, $b$, and the relations become equations in $a$ and $b$. The rule is the same every time: choose names that encode the condition, then apply the sum, product and pair-sum relations and solve the resulting system.