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1: The formula for the difference of two squares is a2 - b2 = (a+b) (a-b). To factor 81x2 - 1 you first write it as the difference of two squares. In the expression 81x2 -1, identify a and b; a=9x and b=1. (9x)2-12 is in the form of a2-b2, so replace a with 9x and replace b with 1 in the formula for the difference of two squares.
A2 - b2 = (a+b)(a-b)
(9x)2 - 12 = (9x+1)(9x-1)
2: Squaring a binomial creates a perfect square trinomial: (a + b)2 (a - b)2 (a + b)2 = a2 + 2ab + b2 (a - b)2 = a2 - 2ab + b2
Factor: x2 + 12x + 36 Solution: Does this fit the pattern of a perfect square trinomial?
Yes. Both x2 and 36 are perfect squares, and 12x is twice the product of x and 6.
Since all signs are positive, the pattern is (a + b)2 = a2 + 2ab + b2. Let a = x and b = 6.
3: 36a2-60a+25. When factoring a trinomial, the first step is to factor out any common factors. The trinomial, 36a2-60a+25, doesn't have any common factors besides 1. The next step is to determine if the trinomial is a perfect square trinomial. A trinomial is a perfect square when two terms, a2 and b2, are squares and the other term is 2*a*b or -2*a*b, this term is twice the product of a and b. 36a2=(6a)2 and 25=(5)2 are perfect squares and -60a=-2*6a*5,36a2-60a+25, is a perfect square trinomial. A perfet square trinomial of the form a2-2ab+b2 is factored as (a-b)2. 36a2-60a+25=(6a)2*5+52= (6a-5)2.