Quadratic Equations
Quadratic Equations  JEE & CBSE
Concepts:
 Quadratic equation:
Equation in the form of $$ ax^2 + bx + c = 0$$, where $$a \neq 0$$
 Standard form:
Rearranged form of the equation as $$ax^2 + bx + c = 0$$
 Roots of quadratic equations:
Values of $$x$$ that make $$ax^2 + bx + c = 0$$ true
 Discriminant (D):
Quantity under the square root in the quadratic formula, $$D = b^2  4ac$$
 Determines nature of roots:
 $$D > 0$$, real and distinct roots
 $$D = 0$$, equal roots
 $$D < 0$$, no real roots (complex and conjugate)
 Quadratic formula: $$ x = \frac{b \pm \sqrt{D}}{2a}$$
Results:

Sum of roots: $$ x_1 + x_2 = \frac{b}{a}$$

Product of roots: $$ x_1 x_2 = \frac{c}{a}$$

Special Cases:

Perfect square: Discriminant is a perfect square, rational equal roots

Incomplete quadratic equation: One variable, forms $$px^2 \pm q =0$$ or $$ax^2 = k, k \neq 0$$

Applications:

Solving quadratic inequalities

Maxima and minima problems

Sums of squares series

Problems involving quadratic relationships.