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Thread: fastest way to factor quadratic

  1. #1
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    fastest way to factor quadratic

    What is the fastest possible way to factor a quadratic? Can someone explain to me how to use the guess and check method, and any limitations it may have (e.g. coefficient in front of $\displaystyle x^2$ must be 1)

    For example how would someone factor:
    $\displaystyle 6x^2+23x+20$
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  2. #2
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    factoered from must be

    $\displaystyle (ax+b)(cx+d) $ where $\displaystyle a,b,c,d \in \Re$

    in your example $\displaystyle 6x^2+23x+20$

    $\displaystyle a\times c = 6$ and $\displaystyle b\times c = 20 $

    now starting guessing and checking:

    1) $\displaystyle a=2, b= -4, c=3, d= -5$ so $\displaystyle (2x-4)(3x-5) $ now expand and check.

    keep guessing until you get the correct answer. Your example has many combinations to try, good luck...
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  3. #3
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    Another thing to think about is the signs. If you take a quadratic $\displaystyle ax^2 \pm bx \pm c$, where a is positive...

    If the quadratic as two plus signs, then the factored binomials both have plus signs:
    $\displaystyle ax^2\;{\color{red}+}\;bx\;{\color{red}+}\;c = (?\;{\color{red}+}\;?)(?\;{\color{red}+}\;?)$

    If the quadratic as a plus sign followed by a minus sign, then the factored binomials both have minus signs:
    $\displaystyle ax^2\;{\color{red}-}\;bx\;{\color{red}+}\;c = (?\;{\color{red}-}\;?)(?\;{\color{red}-}\;?)$

    Otherwise, the factored binomials will have one plus sign and one minus sign:
    $\displaystyle ax^2\;{\color{red}+}\;bx\;{\color{red}-}\;c = (?\;{\color{red}+}\;?)(?\;{\color{red}-}\;?)$
    OR
    $\displaystyle ax^2\;{\color{red}-}\;bx\;{\color{red}+}\;c = (?\;{\color{red}+}\;?)(?\;{\color{red}-}\;?)$

    Remembering this may cut down the number of guess-and-checks you have to try when factoring.


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