Factor by any method. See Examples 1–7. (2y-1)^2-4(2y-1)+4

Verified step by step guidance

1. Recognize that the expression \((2y-1)^2 - 4(2y-1) + 4\) is a quadratic in terms of \(u = 2y-1\).
2. Rewrite the expression as \(u^2 - 4u + 4\).
3. Notice that \(u^2 - 4u + 4\) is a perfect square trinomial.
4. Factor the perfect square trinomial as \((u - 2)^2\).
5. Substitute back \(u = 2y-1\) to get \((2y-1 - 2)^2\) or \((2y-3)^2\).

Key Concepts

Factoring

Factoring is the process of breaking down an expression into simpler components, or factors, that when multiplied together yield the original expression. This is a fundamental skill in algebra, allowing for easier manipulation and solving of equations. Common methods include factoring out the greatest common factor, using the difference of squares, and applying the quadratic formula.

Quadratic Expressions

A quadratic expression is a polynomial of degree two, typically in the form ax^2 + bx + c. In the given question, the expression can be rewritten in terms of a single variable, making it easier to identify its structure. Understanding how to manipulate and factor these expressions is crucial for solving quadratic equations and analyzing their properties.

Completing the Square

Completing the square is a method used to transform a quadratic expression into a perfect square trinomial, which can then be factored easily. This technique involves adjusting the expression by adding and subtracting the same value, allowing for a clearer view of the roots and vertex of the quadratic. It is particularly useful in solving equations and graphing parabolas.