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Thread: [SOLVED] Factor or reverse FOIL help needed

  1. #1
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    [SOLVED] Factor or reverse FOIL help needed

    3x^2-2x-1=0

    I've completely forgotten how to factor more complex quadratic equations such as this. Please help. Thank you.
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  2. #2
    Senior Member Twig's Avatar
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    Hi!

    $\displaystyle 3x^{2}-2x-1 $

    You can solve $\displaystyle 3x^{2}-2x-1=0 $, that is, find the roots of the polynomial. This can be done by using the quadratic formula:


    $\displaystyle x^{2}+px+q=0 \mbox{ , Solution formula: } x = -\frac{p}{2}\pm \sqrt{(\frac{p}{2})^{2}-q} $

    This will give you two roots. Call these roots a and b (assuming here that $\displaystyle a \neq b $ ). So you polynomial can be written

    $\displaystyle (x-a)(x-b) $


    We do the same with your polynomial:

    $\displaystyle 3x^{2}-2x-1=0 \Longleftrightarrow x^{2}-\frac{2}{3}x-\frac{1}{3} =0$

    Now using the quadratic formula we get: $\displaystyle x=\frac{2}{6} \pm \sqrt{\frac{4}{3}+\frac{1}{3}} $ , which gives the two solutions

    $\displaystyle x = 1 \mbox{ and } x = -\frac{1}{3} $

    So we can factor your polynomial as $\displaystyle 3(x-1)(x+\frac{1}{3}) $ , and we multiply it out we indeed get what we started with.
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