%5E2-4(x-2y)%2B4.jpeg "(x-2y)^2-4(x-2y)+4")
Factoring a Quadratic Expression: (x-2y)^2 - 4(x-2y) + 4
This article explores factoring the quadratic expression (x-2y)^2 - 4(x-2y) + 4. We will use a combination of algebraic manipulation and recognizing patterns to simplify the expression.
Recognizing the Pattern
Notice that the expression resembles a perfect square trinomial. A perfect square trinomial is a trinomial that results from squaring a binomial. The general form of a perfect square trinomial is:
(a + b)^2 = a^2 + 2ab + b^2 or (a - b)^2 = a^2 - 2ab + b^2
Let's try to fit our expression into this pattern.
Factoring the Expression
- Identify 'a' and 'b':
- In our expression, a = (x - 2y) and b = 2.
- Verify the pattern:
- (x-2y)^2 = a^2
- -4(x-2y) = -2 * a * b
- 4 = b^2
- Apply the pattern:
- Since the expression matches the pattern of a perfect square trinomial, we can factor it as: (a - b)^2
Solution
Therefore, the factored form of (x-2y)^2 - 4(x-2y) + 4 is (x - 2y - 2)^2.
Conclusion
By recognizing the pattern of a perfect square trinomial, we were able to factor the expression (x-2y)^2 - 4(x-2y) + 4 into (x - 2y - 2)^2. This simplification makes the expression easier to work with in further algebraic manipulations.