Quadratic Equations and Theory of Equations — Formula Sheet

ax^2 + bx + c = 0

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

D = b^2 - 4ac

\alpha + \beta = -\frac{b}{a}

\alpha \beta = \frac{c}{a}

\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta

f(x) = ax^2 + bx + c

x = -\frac{b}{2a}

\frac{4ac - b^2}{4a}

x^n + p_1x^{n-1} + \cdots + p_n = 0

\sum \alpha = -p_1

\sum \alpha\beta = p_2

\alpha\beta\gamma = -p_3

A quadratic equation is of the form $ax^2 + bx + c = 0$, where $a \neq 0$.

Discriminant $D = b^2 - 4ac$ decides the type of roots.

If $D = 0$: two equal real roots (repeated root).

For $ax^2 + bx + c = 0$, sum of roots $\alpha + \beta = -\frac{b}{a}$

If D = 0, expression has same sign as 'a' except at vertex.

The zero-product property states that if $ ab = 0 $, then $ a = 0 $ or $ b = 0 $

For a quadratic equation $ax^2 + bx + c = 0$, the sum of roots $\alpha + \beta = -\frac{b}{a}$ and the product $\alpha\beta = \frac{c}{a}$.