Quadratic equations. Vieta's theorem

A quadratic equation $ax^2 + bx + c = 0$ (with $a \neq 0$) has two roots, which can be found using the quadratic formula or factoring. Vieta's theorem states that if $x_1$ and $x_2$ are the roots, then:

$x_1 + x_2 = -\frac{b}{a}, \quad x_1 x_2 = \frac{c}{a}$

These relationships allow us to solve problems about the roots without explicitly finding them.

Important properties

• The sum and product of the roots depend only on the coefficients.

• If the quadratic can be factored as $a(x - x_1)(x - x_2)$, expanding gives the same coefficients.

• Vieta's theorem can be used to construct quadratic equations from given roots.