Difference of Squares

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Difference of squares (something squared minus something else squared):

` huge color(purple)(a^2-b^2=(a-b)(a+b)) `

Proof of this fact is straightforward.

We just prove it from right to left.

Apply FOIL to `(a-b)(a+b)`:

`(a-b)(a+b)=a*a+a*b+(-b)*a+(-b)*b=a^2+ab-ab-b^2=a^2-b^2`.

This formula can also be derived by applying techniques from Factoring Quadratics note.

Note: formula is valid only for `a^2-b^2`. Sum of squares `a^2+b^2` can't be factored at all.

Example 1. Factor `x^2-9`.

Notice, that `9=3^2`.

Thus, `x^2-9=x^2-3^2=(x-3)(x+3)`.

Answer: `x^2-9=(x-3)(x+3)`.

Of course, there can be more complex expressions.

Example 2. Factor `9y^2-49`.

Notice, that `9y^2=(3y)^2` and `49=7^2`.

Thus, `9y^2-49=(3y)^2-7^2=(3y-7)(3y+7)`.

Answer: `9y^2-49=(3y-7)(3y+7)`.

And even harder...

Example 3. Factor the following: `(x+y)^2-25u^4b^6`.

Notice, that `25u^4b^6=(5u^2b^3)^2`.

Thus, `(x+y)^2-25u^4b^6=(x+y)^2-(5u^2b^3)^2=((x+y)-5u^2b^3)((x+y)+5u^2b^3)`.

Answer: `(x+y)^2-25u^4b^6=(x+y-5u^2b^3)(x+y+5u^2b^3)`.

Now, it is time to exercise.

Exercise 1. Factor the following: `n^2-36`.

Answer: `(n-6)(n+6)`.

Exercise 2. Factor the following: `-1+49x^2`.

Answer: `(7x-1)(7x+1)`. Hint: `-1+49x^2=49x^2-1`.

Exercise 3. Factor `144c^10d^8-(m+n)^2`.

Answer: `(12c^5d^4-m-n)(12c^5d^4+m+n)`.

Exercise 4. Factor `(x+y)^2-(x-y)^2`.

Answer: `4xy`.