Algebraic factorization
1) (x+3)^2 - (x+1)^2
2) (x-4)^2-16(x+2)^2
3) 4(x+1)^2 -(2-x)^2
all these can be solved by using the formula
a^2-b^2=(a+b)*(a-b)
(1)---> ans. is (2x+4)*(2)
(2)---> ans. is (5x+4)*(-3x-4)
(3)---> ans. is (3x+6)*(5x-2)
To solve these algebraic factorization problems, we can use the difference of squares formula, which states that:
a^2 - b^2 = (a + b)(a - b)
Let's apply this formula to the given expressions:
1) (x+3)^2 - (x+1)^2
First, we identify that a = x+3 and b = x+1.
Applying the formula, we have:
(x+3)^2 - (x+1)^2 = [(x+3) + (x+1)][(x+3) - (x+1)]
Simplifying further:
= (2x + 4)(2)
= 4x + 8
Therefore, the factorized form of (x+3)^2 - (x+1)^2 is 4x + 8.
2) (x-4)^2 - 16(x+2)^2
In this expression, a = x-4 and b = 4(x+2).
Using the difference of squares formula, we can write:
(x-4)^2 - 16(x+2)^2 = [(x-4) + 4(x+2)][(x-4) - 4(x+2)]
Expanding and simplifying:
= (x-4 + 4x + 8)(x-4 - 4x - 8)
= (5x + 4)(-3x - 12)
Therefore, the factorized form of (x-4)^2 - 16(x+2)^2 is (5x + 4)(-3x - 12).
3) 4(x+1)^2 - (2-x)^2
Here, a = x+1 and b = (2-x).
Applying the difference of squares formula:
4(x+1)^2 - (2-x)^2 = [4(x+1) + (2-x)][4(x+1) - (2-x)]
Simplifying:
= (4x + 4 + 2 - x)(4x + 4 - 2 + x)
= (3x + 6)(5x + 2)
Therefore, the factorized form of 4(x+1)^2 - (2-x)^2 is (3x + 6)(5x + 2).
By using the difference of squares formula and simplifying the expressions, we were able to obtain the factorized forms of the given expressions.