Using Difference of Squares

Mental Math · Axiom Academy

LESSON Using Difference of Squares Two numbers an equal step either side of a round number? Square the middle, then subtract the step squared. Every pair of numbers has a middle: add them and halve. What makes 48 × 52 worth a second look is that their middle, 50 , is a number whose square you already know. Once you see that, you stop reading the pair as 48 and 52 and start reading it as 50 − 2 and 50 + 2 — the same distance below and above the middle. Call the middle m and the distance d . Every pair built that way multiplies the same way: square the middle, then take off the square of the distance. For 48 × 52 that is 2500 − 4 = 2496 . The identical move handles 53 × 47 , where the middle is again 50 and the distance is 3: 2500 − 9 = 2491 . Read left to right it is ordinary algebra. Read right to left it is a shortcut: a product of two numbers spaced equally around m collapses into one square minus one small square. 48 × 52, rewritten around its middle The smallest version of the same move 31 × 29 is 30 + 1 and 30 − 1, so it is 900 − 1 = 899 — done before you have finished reading the question. 61 × 59 is the same shape one decade up: 3600 − 1 = 3599 . 2. Cut the Square, Slide the Piece This is not a coincidence to memorise — you can watch the reason happen. Start with a square of side m , so its area is m². Cut a small square of side d off one corner and throw it away. What is left is an L-shape of area m² − d² .

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