Final answer:
To simplify the expression (2x-2)^2 - 2(x-3)^2, expand each squared binomial and subtract twice the second expansion from the first expansion to combine like terms. The result is 2x^2 + 4x - 14.
Step-by-step explanation:
To simplify the given expression (2x-2)^2 - 2(x-3)^2, we need to expand each squared binomial and combine like terms where possible. First, consider the square of a binomial which can be expanded as (a-b)^2 = a^2 - 2ab + b^2:
- (2x-2)^2 = (2x)^2 - 2*(2x)*2 + (2)^2 = 4x^2 - 8x + 4,
- (x-3)^2 = x^2 - 2*x*3 + (3)^2 = x^2 - 6x + 9.
Then we subtract twice the second expansion from the first expansion:
4x^2 - 8x + 4 - 2*(x^2 - 6x + 9) = 4x^2 - 8x + 4 - 2x^2 + 12x - 18,
Combining like terms yields:
4x^2 - 8x + 4 - 2x^2 + 12x - 18 = (4x^2 - 2x^2) + (-8x + 12x) + (4 - 18) = 2x^2 + 4x - 14.
The simplified expression is 2x^2 + 4x - 14.