## College Algebra (11th Edition)

$(2y-3)^2$
Let $z=(2y-1).$ Then the given expression, $(2y-1)^2-4(2y-1)+4 ,$ is equivalent to \begin{array}{l}\require{cancel} z^2-4z+4 .\end{array} The two numbers whose product is $ac= 1(4)=4$ and whose sum is $b= -4$ are $\{ -2,-2 \}$. Using these two numbers to decompose the middle term of the expression, $z^2-4z+4 ,$ then \begin{array}{l}\require{cancel} z^2-2z-2z+4 \\\\= (z^2-2z)-(2z-4) \\\\= z(z-2)-2(z-2) \\\\= (z-2)(z-2) \\\\= (z-2)^2 .\end{array} Since $z=(2y-1)$, then the expression $(z-2)^2$ is equivalent to \begin{array}{l}\require{cancel} (2y-1-2)^2 \\\\= (2y-3)^2 .\end{array}