1. Introduction – परिचय

An equation of the form
ax2 + bx + c = 0, where a, b, c are real numbers and a ≠ 0,
is called a Quadratic Equation. The highest power of the variable is 2.

Quadratic equations are used in problems related to area, speed, time,
profit & loss and geometry.

जिस समीकरण में चर की अधिकतम घात 2 होती है, उसे द्विघात समीकरण कहते हैं।
इसका सामान्य रूप है:

ax2 + bx + c = 0, (a ≠ 0)

2. Quadratic Equation – द्विघात समीकरण

English: A quadratic equation is a polynomial equation of degree 2.

Examples:

  • x2 − 5x + 6 = 0
  • 2x2 + 7x − 4 = 0

हिंदी: द्विघात समीकरण वह बहुपद समीकरण है जिसकी घात 2 होती है।

3. Standard Form of a Quadratic Equation – द्विघात समीकरण का मानक रूप

English: The standard form of a quadratic equation is:

ax2 + bx + c = 0

  • a = coefficient of x2
  • b = coefficient of x
  • c = constant term

हिंदी: द्विघात समीकरण का मानक रूप:

ax2 + bx + c = 0

4. Solution by Factorisation Method – गुणनखंड विधि

Steps:

  1. Write the equation in standard form.
  2. Factorise the quadratic expression.
  3. Equate each factor to zero.
  4. Solve for x.

Example:

x2 − 5x + 6 = 0

(x − 2)(x − 3) = 0

x = 2 or x = 3

5. Completing the Square Method – वर्ग पूर्ण करने की विधि

In this method, the quadratic equation is converted into a perfect square.

Example:

x2 + 6x + 5 = 0
x2 + 6x = −5
(x + 3)2 = 4
x = −1 or x = −5

6. Quadratic Formula – द्विघात सूत्र

For the equation ax2 + bx + c = 0, the solution is:

x = (−b ± √(b2 − 4ac)) / 2a

7. Discriminant – विविक्तांक

The expression b2 − 4ac is called the Discriminant.

D = b2 − 4ac

8. Nature of Roots – मूलों का स्वभाव

The nature of roots of a quadratic equation depends on the value of the discriminant.

9. Relationship between Discriminant and Nature of Roots

Discriminant (D)Nature of Roots
D > 0Two distinct real roots
D = 0Two equal real roots
D < 0No real roots

10. Word Problems based on Quadratic Equations – कथनात्मक प्रश्न

English: Word problems involve forming a quadratic equation from given conditions
and solving it.

Example:

The product of two consecutive positive integers is 132.

Let the integers be x and x + 1

x(x + 1) = 132
x2 + x − 132 = 0

हिंदी: कथनात्मक प्रश्नों में दी गई जानकारी से द्विघात समीकरण बनाकर हल किया जाता है।

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