Unit 3 · Topic 04 · Algebra
Meera gave Arjun 60 seconds to factorise x² − 81 before she finished tying her shoelace.
Arjun reached for the (x+a)(x+b) method and started hunting for two numbers that multiply to -81 and add to 0.
Meera finished her laces, glanced at the problem, and said "9 and 9" without touching a pen.
"That's not fair," Arjun said. "You didn't even work it out." Meera pointed at the expression again: x² − 81. "There's no middle term. That's the whole clue. x² is a perfect square, 81 is a perfect square, and there's a minus sign between them. That's the difference-of-squares pattern from the identities topic, backwards."
She wrote it out: a² − b² = (a + b)(a − b). "Here a is x and b is 9, since 9² is 81. So x² − 81 factors straight to (x + 9)(x − 9). No middle term to split, no trial and error."
Arjun checked by expanding (x+9)(x−9): x² − 9x + 9x − 81, and the two middle terms cancelled, leaving x² − 81 exactly. "So recognising the SHAPE saves the whole search."
Meera then wrote x² + 10x + 25 on a fresh line. "This one's a perfect square trinomial. Check: is the first term a square? Yes, x². Is the last term a square? Yes, 25 is 5². Is the middle term twice the product of the two roots? 2 times x times 5 is 10x. It matches, so this is (x + 5)² — nothing to factor by trial, just recognise the pattern."
Arjun tried one himself: 4x² − 12x + 9. First term 4x² is (2x)². Last term 9 is 3². Middle term should be 2 times 2x times 3, which is 12x — and the sign is minus, matching the (a−b)² pattern. "So it's (2x − 3)² he said, and expanded it to check: 4x² − 6x − 6x + 9 = 4x² − 12x + 9. Correct."
"The identities aren't just for expanding any more," Meera said. "Now they're a checklist. No middle term and both ends are squares with a minus between them — difference of squares. Middle term equals twice the square roots of the ends — perfect square trinomial."
Arjun tried to trip her up with x² + 9. "Both squares, but there's a plus, no middle term. Does that factor with real numbers?" Meera shook her head. "No — the difference-of-squares pattern needs a MINUS between the two squares. A sum of two squares like this doesn't factor at all using real numbers."
"So the checklist tells you when NOT to bother too," Arjun said, and started spotting patterns everywhere on the page.
Recognising an identity's SHAPE lets you factorise instantly instead of hunting by trial and error. Two shapes matter most: a² − b² = (a+b)(a−b) — two perfect squares separated by a minus with no middle term — and a² ± 2ab + b² = (a±b)² — a perfect square trinomial, where the first and last terms are squares and the middle term is exactly twice the product of their square roots.
To check for difference of squares: are both terms perfect squares, and is there a minus sign between them with nothing else? To check for a perfect square trinomial: are the first and last terms perfect squares, and does doubling the product of their square roots give exactly the middle term (matching sign)?
A sum of two squares, like x² + 9, does NOT factor using these identities — the pattern strictly needs a minus sign, and there is no real-number factorisation for a genuine sum of two squares.
Spotting the shape first, before trying anything else, is much faster than guessing pairs of numbers — the whole point of learning identities both forwards (expanding) and backwards (factorising).
Factorise x² − 49, x² − 16, and 9x² − 25 using the difference-of-squares shape, checking each by expanding back.
Take x² + 6x + 9 and 4x² − 4x + 1, confirm each is a genuine perfect square trinomial by checking all three parts of the shape, then write each as a single squared bracket.
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