Vieta's Theorem (Vieta's Formulas)
🎯 Big Idea
Vieta's Theorem gives us a powerful shortcut: we can find the sum and product of the roots of a quadratic equation without actually solving it!
📐 The Setup
Consider a general quadratic equation:
$$ ax^2 + bx + c = 0 $$
where $a \neq 0$, and let $x_1$ and $x_2$ be its two roots (solutions).
⭐ Vieta's Theorem — The Formulas
Sum of Roots
$$ x_1 + x_2 = -\frac{b}{a} $$
Product of Roots
$$ x_1 \cdot x_2 = \frac{c}{a} $$
In words: The sum of the roots equals the negative of the coefficient $b$ divided by $a$, and the product of the roots equals the constant term $c$ divided by $a$.