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Factor. $\newline$ $9p^2 - 12p + 4$

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Factor quadratics: special cases

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Q. Factor.

\newline

9p^2 - 12p + 4

Identify Perfect Square Trinomial: Identify if the quadratic can be factored using the perfect square trinomial formula. $\newline$ A perfect square trinomial is in the form $(a^2 \pm 2ab + b^2)$ which factors to $(a \pm b)^2$ . $\newline$ Check if $9p^2 - 12p + 4$ is a perfect square trinomial. $\newline$ $9p^2$ is a perfect square, as $(3p)^2 = 9p^2$ . $\newline$ $4$ is a perfect square, as $2^2 = 4$ . $\newline$ The middle term, $-12p$ , is twice the product of the square roots of the first and last terms, as $2 \times 3p \times 2 = 12p$ . $\newline$ Thus, $9p^2 - 12p + 4$ is a perfect square trinomial.
Check for Perfect Square Trinomial: Factor the perfect square trinomial using the formula $(a^2 - 2ab + b^2) = (a - b)^2$ . Here, $a = 3p$ and $b = 2$ . So, $(3p)^2 - 2 \times 3p \times 2 + 2^2 = (3p - 2)^2$ . Therefore, the factored form of $9p^2 - 12p + 4$ is $(3p - 2)^2$ .

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