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Factor. $\newline$ $25w^2 - 30w + 9$

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Factor quadratics: special cases

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Q. Factor.

\newline

25w^2 - 30w + 9

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 $25w^2 - 30w + 9$ fits this pattern. $\newline$ $25w^2$ is a perfect square, as it is $(5w)^2$ . $\newline$ $9$ is a perfect square, as it is $3^2$ . $\newline$ The middle term, $-30w$ , 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. $\newline$ $2 \times 5w \times 3 = 30w$ , but we have $-30w$ , so it still fits the pattern with a negative sign.
Write as Perfect Square: Write the expression as a perfect square trinomial. $\newline$ Since we have established that $25w^2 - 30w + 9$ is a perfect square trinomial, we can write it as: $\newline$ $(5w)^2 - 2 \times 5w \times 3 + 3^2$ $\newline$ This confirms that the expression is indeed a perfect square trinomial.
Factor Trinomial: Factor the perfect square trinomial. $\newline$ Using the formula $(a^2 - 2ab + b^2) = (a - b)^2$ , we can factor the expression as: $\newline$ $(5w - 3)^2$ $\newline$ This is the factored form of the quadratic expression.

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