Two squares make a pattern

The idea

Adding the expansions of (a + b)² and (a – b)² gives 2(a² + b²) = (a + b)² + (a – b)².

A pattern of squares

2(2² + 1²) = 3² + 1² 2(3² + 1²) = 4² + 2² 2(6² + 5²) = 11² + 1² 2(5² + 3²) = 8² + 2²

Notice: 2(5² + 6²) = (6 + 5)² + (6 − 5)². The new numbers are the sum and the difference.

Will this work for any two numbers?

• Yes, for any two numbers
• Only when the numbers are close together
• No, only for these four lines

Try pairs, then add the columns

Pick two numbers a and b. Both sides match. Then add the two expansions, column by column, to see why.

The middle terms cancel

2(a² + b²) = (a + b)² + (a − b)² It comes from adding the expansions of (a + b)² and (a − b)².

(a + b)² + (a − b)² = (a² + 2ab + b²) + (a² − 2ab + b²)

Add the like terms: a² + a² = 2a² b² + b² = 2b² 2ab − 2ab = 0 So 2(a² + b²) = (a + b)² + (a − b)².

Notes

2(a² + b²) = (a + b)² + (a − b)² Add the two expansions, and the middle terms 2ab and −2ab cancel.

Check yourself

Complete the pattern: 2(9² + 4²) = 13² + □². What number goes in the box?

Answer: 5

(9 + 4)² + (9 − 4)² = 13² + 5² = 169 + 25 = 194 = 2(81 + 16).

5² + 1² = 2(a² + b²). Read the identity backwards. Which a and b fit?

Without squaring the brackets, find (13 + 5)² + (13 − 5)². Use 2(a² + b²).

Answer: 388

2(13² + 5²) = 2 × (169 + 25) = 2 × 194 = 388. Check: 18² + 8² = 324 + 64 = 388.

Which two terms cancel when we add (a + b)² and (a − b)²?

• a = 3, b = 2 — correct. Yes! 3 + 2 = 5 and 3 − 2 = 1. Check: 2(9 + 4) = 26 = 25 + 1.
• a = 4, b = 1. 4 + 1 = 5 but 4 − 1 = 3, not 1. And 2(16 + 1) = 34, not 26.
• a = 5, b = 1. 5 + 1 = 6, not 5. And 2(25 + 1) = 52, not 26.
• a² and a². Both are +a². They add up to 2a². They do not cancel.
• b² and b². Both are +b². They add up to 2b². They do not cancel.
• 2ab and −2ab — correct. Yes! 2ab − 2ab = 0. The middle terms vanish.