Sum and product of the roots

Vieta’s formulas express the sum and the product of the two solutions in terms of the coefficients $a$, $b$, $c$ alone: there is no need to know the solutions in order to compute them.

Property — Vieta's formulas

If $x_A$ and $x_B$ are the solutions of $ax^2+bx+c=0$ ($a\neq 0$), then: $x_A+x_B = -\frac{b}{a}, \qquad x_A\cdot x_B = \frac{c}{a}.$

These two relations are valuable in many contexts: they let you quickly check the solutions you have found, reconstruct an equation from its roots and, as we shall see, solve symmetric systems.

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Topics: Second-degree equations
Concepts: Second-degree equation · Sum and product of the roots
Methods: Sum and product of the roots
Skills: Using formulas
People: François Viète (Vieta)