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.
What a learner can do afterwards
- 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
1 · Read
In the last stop you hunted 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. One square, one subtraction, no 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.
2 · Watch
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Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
24 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.