Quadratic Equations

Quadratic equations form the backbone of JEE algebra, contributing 1-2 questions annually. The chapter covers the standard quadratic $ax^{2}$ + bx + c = 0, nature of roots, relationships between roots and coefficients, and transformations of equations.

Standard Form and Discriminant: For $ax^{2}$ + bx + c = 0 (a not equal to 0), the roots are x = (-b ± $\sqrt{D}$)/(2a) where D = $b^{2}$ - 4ac is the discriminant. D > 0 gives two distinct real roots, D = 0 gives equal real roots, and D < 0 gives complex conjugate roots. For real coefficients, complex roots always occur in conjugate pairs.

Vieta's Formulas (Sum and Product of Roots): If $\alpha$ and $\beta$ are roots of $ax^{2}$ + bx + c = 0, then $\alpha$ + $\beta$ = -b/a and $\alpha$ * $\beta$ = c/a.
These allow computing symmetric functions of roots without finding the roots themselves: $\alpha$² + $\beta$² = ($\alpha$+$\beta$)² - 2*$\alpha$*$\beta$, and |$\alpha$ - $\beta$| = $\sqrt{D}$/|a|.

Nature of Roots for Real Coefficients: Beyond the discriminant, the sign analysis of f(x) = $ax^{2}$ + bx + c provides powerful tools. If f(k) < 0 for some k and a > 0, then k lies between the roots. The vertex is at x = -b/(2a) with value f(-b/(2a)) = -D/(4a).

Quadratic Function and Graph: y = $ax^{2}$ + bx + c is a parabola opening upward if a > 0, downward if a < 0. The minimum/maximum value is -D/(4a) at x = -b/(2a). The sign of the expression $ax^{2}$ + bx + c depends on the sign of a and the discriminant.

Sign of Quadratic Expression: If a > 0 and D < 0, then $ax^{2}$ + bx + c > 0 for all real x. If a < 0 and D < 0, then $ax^{2}$ + bx + c < 0 for all real x. If D >= 0, the expression changes sign at its roots.

Common Root and Condition: Two quadratics $a_{1x}^{2}$ + $b_{1x}$ + $c_{1}$ = 0 and $a_{2x}^{2}$ + $b_{2x}$ + $c_{2}$ = 0 have a common root if ($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}$).
Both roots common iff $\frac{a_{1}}{a_{2}}$ = $\frac{b_{1}}{b_{2}}$ = $\frac{c_{1}}{c_{2}}$.

Equations Reducible to Quadratics: Biquadratic ($ax^{4}$ + $bx^{2}$ + c = 0 via t = $x^{2}$), reciprocal equations, and equations involving sqrt expressions can often be converted to quadratic form with appropriate substitution.

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