Squares (All Methods)
Master all techniques to square any number.
Core Concepts
We have learned several methods for squaring numbers. Now it's time to choose the best tool for the job!
1. Base Method
Best for numbers near 10, 50, 100, 1000.
96² (Base 100): 96-4 / 4² = 9216.
52² (Base 50): 25+2 / 2² = 2704.
2. Ends in 5
Best for numbers ending in 5.
35²: 3×4 / 25 = 1225.
3. Duplex Method (General)
Works for ANY 2-digit number (ab)².
Formula: a² | 2ab | b²
31²: 3² | 2×3×1 | 1² = 9 | 6 | 1 = 961.
Solved Examples
Example 1: 106² (Base Method)
Question: Calculate 106².
• Base 100. Surplus is +6.
• 106 + 6 = 112.
• 6² = 36.
• Answer: 11236.
Example 2: 42² (Duplex Method)
Question: Calculate 42².
• a=4, b=2.
• 4² | 2×4×2 | 2²
• 16 | 16 | 4
• Carry over the 1 from 16: (16+1) | 6 | 4.
• Answer: 1764.
Example 3: 85² (Ends in 5)
Question: Calculate 85².
• First part: 8 × (8+1) = 8 × 9 = 72.
• Last part: 25.
• Answer: 7225.
