rag/knowledge_base/quadratic_equations.md

Quadratic Equations

Standard Form

$ax^2 + bx + c = 0$ where $a \neq 0$

Quadratic Formula

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

Discriminant

$D = b^2 - 4ac$

• $D > 0$: Two distinct real roots
• $D = 0$: Two equal real roots (repeated root)
• $D < 0$: Two complex conjugate roots

Vieta's Formulas

For roots $\alpha, \beta$:

• Sum of roots: $\alpha + \beta = -\frac{b}{a}$
• Product of roots: $\alpha \cdot \beta = \frac{c}{a}$

Common Transformations

• If roots are $\alpha, \beta$, equation with roots $\frac{1}{\alpha}, \frac{1}{\beta}$: $cx^2 + bx + a = 0$
• Equation with roots $\alpha^2, \beta^2$: replace $x$ with $\sqrt{x}$ and simplify
• Equation with roots $\alpha+k, \beta+k$: replace $x$ with $x-k$

Nature of Roots (JEE Important)

• Both roots positive: $D \geq 0$, $\alpha + \beta > 0$, $\alpha\beta > 0$
• Both roots negative: $D \geq 0$, $\alpha + \beta < 0$, $\alpha\beta > 0$
• Roots of opposite signs: $\alpha\beta < 0$
• Roots lie in interval $(k, l)$: $af(k) > 0$, $af(l) > 0$, $k < -b/2a < l$, $D \geq 0$