having trouble factoring this particular problem
x^2-12x+36-49y2
factored by grouping
x(x-12) (7y-6) (7y+6)
the answer in the book has
(x-6+7y)(x-6-7y)
confused.
first, notice that the first three terms, x^2-12x+36, is a perfect square, and can be factored into (x-6)(x-6),, now rewriting the expression, it becomes:
(x-6)^2 - 49y^2
(x-6)^2 - (7y)^2
this looks like difference of two squares,, let a = x-6 and b=7y, thus:
a^2 - b^2
(a-b)(a+b)
(x-6-7y)(x-6+7y)
so there,, :)
Gotta remember those damn steps. I immediately assumed it was four terms so I factor by grouping. I forgot that you first look for common factor, then squares etc.
Thanks.
To factor the expression x^2-12x+36-49y^2 using grouping, follow these steps:
Step 1: Group the terms.
The expression can be rewritten as:
(x^2 - 12x) + (36 - 49y^2)
Step 2: Factor out common terms.
In the first group, factor out the common term 'x':
x(x - 12) + (36 - 49y^2)
Step 3: Look for a difference of squares.
In the second group, notice that 36 can be written as 6^2 and 49y^2 as (7y)^2:
x(x - 12) + (6^2 - (7y)^2)
Step 4: Apply the difference of squares formula.
The difference of squares formula states that a^2 - b^2 = (a + b)(a - b). In this case, let a = 6 and b = 7y:
x(x - 12) + (6 + 7y)(6 - 7y)
So, the factored form of the expression x^2-12x+36-49y^2 is x(x - 12) + (6 + 7y)(6 - 7y).
However, it seems that the answer in the book has a slightly different form. Let's simplify it to see if it's equivalent to our factored expression:
Step 5: Simplify the answer in the book.
(x-6+7y)(x-6-7y)
Using the distributive property, we can expand this expression:
= x(x-6-7y) - 6(x-6-7y) + 7y(x-6-7y)
Simplifying further, note that (x-6-7y) appears in each term:
= x(x-6-7y) - 6(x-6-7y) + 7y(x-6-7y)
= (x-6-7y)(x) - (x-6-7y)(6) + (x-6-7y)(7y)
Combining like terms, we get:
= x^2 - 6x - 7xy - 6x + 36 + 42y - 7xy + 42y - 49y^2
= x^2 - 12x - 14xy + 36 + 84y - 49y^2
Comparing this simplified expression with our original factored form, we can see that they are indeed equivalent.
So, both x(x - 12) + (6 + 7y)(6 - 7y) and (x-6+7y)(x-6-7y) represent the same factored form of x^2-12x+36-49y^2. The difference lies in the order and presentation of the factors.