A quadratic equation $ax^2$ + bx + c = 0 (a != 0) is solved using the quadratic formula x = (-b +/- sqrt($b^2$ - 4ac))/(2a). The discriminant D = $b^2$ - 4ac determines root nature: D > 0 yields two distinct real roots (rational if D is a perfect square with rational coefficients), D = 0 yields repeated roots, and D < 0 yields complex conjugate roots. For polynomials with real coefficients, complex roots and irrational roots (of the form p +/- sqrt(q)) always appear in conjugate pairs.

Vieta's formulas connect roots to coefficients: sum of roots = -b/a, product = $\frac{c}{a}$. These enable computation of symmetric functions like $alpha^2$ + $beta^2$ = (alpha+beta)^2 - 2alphabeta without finding individual roots. The recurrence $S_n$ = (-b/a)*S_(n-1) - $\frac{c}{a}$*S_(n-2) computes higher power sums efficiently.

The quadratic function f(x) = $ax^2$ + bx + c represents a parabola with vertex at (-$\frac{b}{2a}$, -$\frac{D}{4a}$). For a > 0, the minimum is -$\frac{D}{4a}$; for a < 0, the maximum is -$\frac{D}{4a}$. Sign analysis of $ax^2$ + bx + c uses: if a > 0 and D < 0, expression is positive for all real x.

Location of roots problems require three conditions: (1) discriminant condition D >= 0, (2) function value conditions f(k) > 0 or f(k) < 0 at boundary points, and (3) vertex position conditions. For both roots in interval (p, q): D >= 0, af(p) > 0, af(q) > 0, p < -$\frac{b}{2a}$ < q.

The common root condition for $a_1$$x^2$ + $b_1$x + $c_1$ = 0 and $a_2$$x^2$ + $b_2$x + $c_2$ = 0 is ($c_1$$a_2$ - $c_2$$a_1$)^2 = ($a_1$$b_2$ - $a_2$$b_1$)($b_1$$c_2$ - $b_2$$c_1$). Equations reducible to quadratics include biquadratics (substitution t = $x^2$), reciprocal equations (t = x + 1/x), and equations with sqrt terms requiring squaring.