Difference of Two Squares and Perfect Square Trinomials
Recognise the two quadratics that factorise on sight: a² - b² splitting into (a + b)(a - b), and a perfect square trinomial folding back into one bracket squared.
Ce que l'enfant saura faire ensuite
- Factorise x² - 49 and 9x² - 25 as a difference of two squares
- Spot that x² + 10x + 25 is (x + 5)² from the halved middle coefficient
- Use the difference of two squares to work out 51 × 49 mentally
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Factorising a quadratic usually means hunting for a pair that multiplies to c and adds to b. Two shapes skip the hunt entirely. One is a difference of two squares: two perfect squares joined by a minus. The other is a perfect square trinomial: one bracket, squared. Spotting them saves time, and they keep reappearing, from equations to mental maths.
How to spot a difference of two squares: exactly two terms, each a perfect square, joined by a minus. x squared is a square and 49 is 7 squared, so x squared - 49 qualifies. But x squared + 49 does not, because the plus sign keeps the middle terms alive when you expand.
Factorise x squared - 49. The two squares are x squared and 7 squared, so the brackets are (x + 7)(x - 7). Check by expanding: the outer and inner terms give -7x and +7x, which cancel, and 7 times -7 gives -49. We land exactly on the original.
Now the x squared term carries a coefficient. Factorise 9x squared - 25. The first square is (3x) squared, since 3x times 3x gives 9x squared. The second square is 5 squared. So the brackets are (3x + 5)(3x - 5). Check: the middle terms -15x and +15x cancel, and 5 times -5 gives -25.
A perfect square trinomial folds back into one bracket squared. Expand (x + 5) squared: x squared, then 2 times 5x, then 25, which is x squared + 10x + 25. Running it backward, halve the middle coefficient: 10 over 2 gives 5. Check that 5 squared is 25, and the trinomial is (x + 5) squared.
The same identity does fast number work. 51 times 49 looks awkward, but 51 is 50 + 1 and 49 is 50 - 1. So 51 times 49 = (50 + 1)(50 - 1) = 50 squared - 1 squared = 2500 - 1 = 2499. That takes one square and one subtraction instead of long multiplication.
A difference of two squares, a squared minus b squared, splits into (a + b)(a - b); a perfect square trinomial with middle term 2ab folds into (a + b) squared.
Readings 'Two shapes that factorise on sight', 'The difference of two squares', 'Perfect squares and mental maths' and 'Spot the minus sign' are original to this dataset. All 6 extra questions are original to this dataset.
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