# Finding Squares Using a Known Square: A Simple Formula Explained

Discussion in 'Number Theory' started by VirajAdani, Jul 22, 2024.

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Have you ever wondered how to quickly calculate the square of a number that's near a known square? There's a neat formula that can help with this, using the distance between the number you want to square and a known square.

Formula Explanation:

(Known square) ± {(target number) * (distance * 2) ± (distance²)}

Let's break this down with some examples to make it clear:

Example 1: Calculating 77² when 80² = 6400

Known square: (80² = 6400 )

Target number: ( 77)

Distance: ( 80 - 77 = 3)

Using the formula:

6400 - {77 × (3×2) + (3²)}

6400 − (77×6+9) = 6400−465 = 5929

Therefore, (77² = 5929 ).

Example 2: Calculating 67² when 65²= 4225

Known square: 65² = 4225

Target number: 67

Distance: 67 - 65 = 2

Using the formula:

4225 + {67 × (2×2) - (2²)}

4225 + (67×4−4) = 4225 + 264 = 4489

Therefore, 67² = 4489.

Example 3: Calculating 248² and 253² when 250² = 62500

Known square: 250² = 62500

For 248:

Target number: 248

Distance: 250 - 248 = 2

Using the formula:

62500 - {248 × (2×2) + (2²)}

62500 − (248×4+4) = 62500 − 996 = 61504

Therefore, 248² = 61504.

For 253:

Target number: 253

Distance: 253 - 250 = 3

Using the formula:

62500 + {253 × (3×2) - (3²)}

62500 + (253×6−9) = 62500 + 1509 = 64009

Therefore, 253² = 64009.

Conclusion:
This formula simplifies finding squares of numbers close to known squares by leveraging the difference (or distance) between the numbers and their nearest squares. It involves straightforward arithmetic operations (multiplication, addition, subtraction) that can be easily calculated mentally or with a simple calculator.

Give it a try with other numbers and known squares to get comfortable with the method. It's a handy trick for quick mental math calculations!