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.