Difference of two squares
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Factor x^2 - 36.
Common mix-up
In this example, a student wrote x^2 - 36 = (x - 6)(x - 6).
The student wrote (x - 6)(x - 6), which expands to x^2 - 12x + 36, not the x^2 - 36 they started with.
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See why this works
Why is x^2 - 36 not (x - 6)(x - 6)?
x^2 - 36 looks like an ordinary binomial, but it carries a special pattern. A difference of two squares always factors into a sum times a difference.
a^2 - b^2 = (a - b)(a + b).
A difference of two squares means exactly what it says: a subtraction, a perfect square, and another perfect square. In x^2 - 36, the first term is x squared and the second is 6 squared, and a minus sign sits between them. Anything written as a^2 - b^2 follows one fixed factoring pattern.
The pattern is a^2 - b^2 = (a - b)(a + b). Multiply the factors back and the middle terms vanish: (a - b)(a + b) = a^2 + ab - ab - b^2 = a^2 - b^2. The opposite signs are the whole trick, because the two middle products add to zero and only the squares survive.
So x^2 - 36 = (x - 6)(x + 6). Compare it with (x - 6)(x - 6), which is what a student might write by symmetry. That product is x^2 - 12x + 36, a perfect square with a middle term, and it does not match the original binomial at all.
The pattern only fits a subtraction of squares. x^2 + 36 cannot be factored this way, because a sum of squares keeps a middle term no matter which signs you try. When you see a minus and two perfect squares, use (a - b)(a + b) instantly; FOIL is your proof.
Keep this idea
A difference of two squares factors as (a - b)(a + b). One sign plus, one minus, and the middle terms cancel.
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