Factor. 4y^2 - 4y + 1 ______

Check Perfect Square Trinomial: Determine if the quadratic can be factored using the perfect square trinomial formula. A perfect square trinomial is in the form (a^2 ± 2ab + b^2) = (a ± b)^2. We need to check if 4y^2 - 4y + 1 fits this pattern. 4y^2 can be written as (2y)^2, and 1 can be written as (1)^2. The middle term, -4y, should be equal to 2*(2y)*(1) for it to be a perfect square trinomial. Let's check: 2*(2y)*(1) = 4y, which is the negative of the middle term in our expression. So, 4y^2 - 4y + 1 is a perfect square trinomial.Factor Perfect Square Trinomial: Factor the perfect square trinomial. Since we have established that 4y^2 - 4y + 1 is a perfect square trinomial, we can write it as (2y - 1)^2. This is because (2y)^2 gives us the first term 4y^2, -2*(2y)*(1) gives us the middle term -4y, and (1)^2 gives us the last term +1. Therefore, the factored form of 4y^2 - 4y + 1 is (2y - 1)(2y - 1) or (2y - 1)^2.

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Algebra 1

Factor quadratics: special cases

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Q. Factor.

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4y^2 - 4y + 1

Check Perfect Square Trinomial: Determine if the quadratic can be factored using the perfect square trinomial formula. $\newline$ A perfect square trinomial is in the form $(a^2 \pm 2ab + b^2) = (a \pm b)^2$ . We need to check if $4y^2 - 4y + 1$ fits this pattern. $\newline$ $4y^2$ can be written as $(2y)^2$ , and $1$ can be written as $(1)^2$ . The middle term, $-4y$ , should be equal to $2\cdot(2y)\cdot(1)$ for it to be a perfect square trinomial. $\newline$ Let's check: $2\cdot(2y)\cdot(1) = 4y$ , which is the negative of the middle term in our expression. $\newline$ So, $4y^2 - 4y + 1$ is a perfect square trinomial.
Factor Perfect Square Trinomial: Factor the perfect square trinomial. $\newline$ Since we have established that $4y^2 - 4y + 1$ is a perfect square trinomial, we can write it as $(2y - 1)^2$ . $\newline$ This is because $(2y)^2$ gives us the first term $4y^2$ , $-2\times(2y)\times(1)$ gives us the middle term $-4y$ , and $(1)^2$ gives us the last term $+1$ . $\newline$ Therefore, the factored form of $4y^2 - 4y + 1$ is $(2y - 1)(2y - 1)$ or $(2y - 1)^2$ .