Class - X, Quadratic Equations

• Introduction:

The general form of a quadratic equation is

• Sum of roots = –b/a
• Product of roots = c/a
• If roots of a quadratic equation are given, then the quadratic equation can be represented as:

                   x² – (sum of the roots)x + product of the roots = 0

Method of solving a quadratic equation:
(i) Factorisation method

(ii) By the method of completing the square
(iii) By quadratic formula (Shreedhar Acharya)

(i) Factorisation method:

Example:

(ii) By the method of completing the square:

Example: 

(iii) By quadratic formula / Shreedhar Acharya formula:

Example: 

Nature of roots:

If ax² + bx + c = 0, a ≠ 0 be a quadratic equation, then b² — 4ac is known as discriminant and is denoted by D.

i.e. D = b² — 4ac

Case I: If D > 0

Then the roots of the equation are real and unequal.

Note 1: If D > 0 and D is a perfect square then the roots are real, unequal and rational if a, b, c are rational numbers.

Note 2: If D > 0 and D is a non-perfect square then the roots are real, unequal and irrational.

Case II: If D = 0

Then the roots of the equation are real and equal.

Case III: If D < 0

Then the equation have no real roots.