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Factor. $\newline$ $4p^2 - 4p + 1$

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Factor quadratics: special cases

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Q. Factor.

\newline

4p^2 - 4p + 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 $4p^2 - 4p + 1$ fits this pattern. $\newline$ $4p^2$ can be written as $(2p)^2$ , and $1$ can be written as $(1)^2$ . The middle term, $-4p$ , should be equal to $2$ times the product of the square roots of the first and last terms if it is a perfect square trinomial. $\newline$ Let's check: $2 \times (2p) \times (1) = 4p$ , but we have $-4p$ , so it fits the pattern with a negative middle term.
Write as Perfect Square Trinomial: Write the expression as a perfect square trinomial. $\newline$ The expression $4p^2 - 4p + 1$ can be written as $(2p)^2 - 2\times(2p)\times(1) + (1)^2$ . $\newline$ This matches the perfect square trinomial formula $(a - b)^2 = a^2 - 2ab + b^2$ , where $a = 2p$ and $b = 1$ .
Factor Perfect Square Trinomial: Factor the perfect square trinomial. $\newline$ Using the formula $(a - b)^2$ , we can write the factored form of $4p^2 - 4p + 1$ as $(2p - 1)^2$ .

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