Unit 3 · Topic 01 · Algebra
Meera squared 103 in her head in four seconds. Arjun reached for a pen, multiplied 103 by 103 the long way, and got there two minutes later.
They got the same answer. Meera hadn't used a calculator, and she hadn't multiplied anything three-digit by three-digit.
Arjun wanted to know what she'd actually done.
Meera and Arjun were doing warm-up sums before their maths olympiad practice. The sheet asked for 103 squared. Arjun wrote 103 under 103 and started the long multiplication, carrying digits, lining up columns. Meera wrote one line: 100 squared plus two times 100 times 3, plus 3 squared, and had 10,609 before Arjun had finished his second row.
"That's not how you square a number," Arjun said, checking his own working. He got 10,609 too. "Okay, it's how you square a number. How did you know to do that?"
Meera explained that she hadn't squared 103 directly at all. She'd rewritten it as (100 + 3) and squared the sum instead, using a shortcut: (a + b)² always equals a² + 2ab + b². She'd picked a = 100 and b = 3 because both were easy to work with.
Arjun tried it on 98 squared, using a = 100 and b = -2, since 98 is 100 minus 2. He got 100² + 2(100)(-2) + (-2)² = 10000 - 400 + 4 = 9604. He checked it the long way. It matched.
"Why does this even work?" he asked. Meera drew a square of side (a + b) split into four pieces: an a-by-a square, a b-by-b square, and two a-by-b rectangles. "Add up the areas of all four pieces," she said. "a² plus b² plus ab plus ab. That's a² + 2ab + b². The identity isn't a trick — it's just the area of a square, split up."
Arjun tested the sister identity next, (a - b)² = a² - 2ab + b², on 97 squared using a = 100, b = 3. He got 10000 - 600 + 9 = 9409, and it checked out too. Then Meera showed him the last one: (a + b)(a - b) = a² - b², useful for something like 45 × 55, which is (50 + 5)(50 - 5) = 2500 - 25 = 2475.
By the end of practice, Arjun could square three-digit numbers as fast as Meera could. "It's not that you're faster at arithmetic," he said. "You just never do the hard arithmetic in the first place."
"That's the whole point of an identity," said Meera. "It's always true, so you get to choose the easiest path to the same answer."
An algebraic identity is an equation that is true for every value of the letters in it, not just some values — unlike an equation you solve for one answer. The three identities Meera used are the standard ones, worth memorising as shapes, not just formulas: (a + b)² = a² + 2ab + b², (a − b)² = a² − 2ab + b², and (a + b)(a − b) = a² − b².
A fourth pattern, (x + a)(x + b) = x² + (a + b)x + ab, handles multiplying two brackets that share the same first letter but different added numbers.
These identities turn a hard multiplication into an easy one by picking a and b to be round, friendly numbers — a strategy, not a coincidence. They also work in reverse: given a² + 2ab + b², you can instantly write it as (a + b)².
Without a calculator, use an identity to compute 52² and 48 × 52. Then check both on a calculator.
Cut a square of paper into four pieces the way the story's diagram does, using any two side lengths you choose with a ruler, and label each piece's area to see a² + 2ab + b² for yourself.
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