Factoring perfect square trinomials

1). X^2 - 10x + 25 - y^2


2). X^2 - 18x + 81 - y^2

Hint:

1) x^2 - 10x + 25 - y^2 = (x2 - 2*5*x + 52 ) - y^2

and

2) x^2 - 18x + 81 - y^2 = (x^2 - 2*9*x + 92 ) - y^2

and continue......
 
Before trying to factor, presumably you learned to multiply:
\(\displaystyle (x+ a)^2= (x+ a)(x+ a)= (x)(x)+ (x)(a)+ (a)(x)+ (a)(a)= x^2+ 2ax+ a^2\).

It is that "2" in the "mixed" term that is crucial! If you have, say \(\displaystyle x^2+ 12x+ 36- y^2\) you should immediately look at that "12x" and think "half of 12 is 6 and \(\displaystyle 6^2= 36\) so the first three terms are a "perfect square trinomial": \(\displaystyle (x+ 6)^2\).

Now, do you know how to factor \(\displaystyle a^2- b^2\)? (The multiplication to look at is (a- b)(a+ b).)
 
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