Description / Overview

In this comprehensive overview of Quadratic Surd (দ্বিঘাত করণী) from the Class 10 West Bengal Board syllabus, students will gain a strong grasp of what surds are, how to convert them into quadratic surds, and apply standard algebraic operations. Clear definitions, step‑by‑step methods, and illustrative examples make the content engaging and accessible while preserving syllabus fidelity.

Structure

1. Introduction to Surds and Quadratic Surds

Surds are irrational numbers expressed in root form that cannot be simplified into rational numbers (e.g., √2, √5). A quadratic surd (দ্বিঘাত করণী) specifically refers to expressions of the form √a where a is not a perfect square. These are fundamental in algebra and essential in Class 10 West Bengal mathematics.

2. Simplifying Quadratic Surds (দ্বিঘাত করণীকে সরলীকরণ)

Key Process: Break the number under the radical into factors, extract perfect squares.

Example: Simplify √72.

• Factor inside: 72 = 36 × 2

• √72 = √(36 × 2) = √36 × √2 = 6√2
  This shows how to reduce a surd to simplest form—crucial for clarity and manipulation.

Further Example: Simplify √50:

• 50 = 25 × 2 → √50 = 5√2

These steps align with the WB Board guidelines on radical simplification.

3. Addition and Subtraction of Like Surds

Only like surds (same radical part) can be directly added or subtracted.

Example: Simplify 3√2 + 5√2 – 2√3.

• Combine like surds: (3√2 + 5√2) = 8√2

• Cannot combine √3, so final = 8√2 – 2√3

Example: Simplify 7√5 – 2√5 + 4√20 → first simplify √20 = 2√5 → so 4×2√5 = 8√5 → total = (7√5 – 2√5 + 8√5) = 13√5.

4. Multiplication of Quadratic Surds

Multiply the numerical and radical parts separately.

Example: Multiply √2 × √8.

• √2 × √8 = √(2 × 8) = √16 = 4

Example: Multiply 3√3 × 2√12.

• First simplify inside: √12 = 2√3

• So the product becomes 3√3 × 2 × 2√3 = (3×2×2) × (√3 × √3) = 12 × 3 = 36

This method aligns with WB Board’s emphasis on step‑wise, structured solutions.

5. Rationalising the Denominator

WB Board requires rationalising when a surd appears in the denominator.

Basic Example: Simplify 5 / √2.

• Multiply numerator and denominator by √2:
  5√2 / (√2 × √2) = 5√2 / 2

More Complex Example: Rationalise 3 / (2 + √3).

• Multiply numerator and denominator by conjugate (2 − √3):

3(2−3​)(2+ √3) (2- √3) = 3(2−3​)4-3 = 3(2−3​)

Final simplified form: 6-33

Rationalising demonstrates how to remove irrational parts from denominators, which the syllabus expects students to handle with ease.

6. Solving Quadratic Equations Leading to Surds

When quadratic equations yield irrational solutions, surds emerge naturally.

Example: Solve x² – 5x + 6 = 0 (roots: x = 2, 3 → rational).
Better example: Solve x² – 2x – 3 = 0:

• Use quadratic formula: 

X = 2(-2)2 – 4(1)(-3)2 = 24+122 = 2162 = 242

 

• Roots: x = 3 or x = −1 (still rational—choose a non‑perfect square).
  Use x² – 2x – 1 = 0:

• Discriminant = 4 + 4 = 8 → x = [2 ± √8]/2 = [2 ± 2√2]/2 = 1 ± √2 → surd form.