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Math Help - relation of delta and factorization?

  1. #1
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    relation of delta and factorization?

    for a quadratic "expression"

    i.e. ax^2 + bx + c

    with any real and integral value of a, b and c

    how to show that if delta (b square minus 4ac) is a perfect square

    then such a quadratic expression can be factorized by cross method?

    into something like (mx-n)(px-q) where m,n,p,q are integers.

    I had been thinking about it for a whole night.

    Hope someone can help
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  2. #2
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    Quote Originally Posted by kenny1999 View Post
    for a quadratic "expression"

    i.e. ax^2 + bx + c

    with any real and integral value of a, b and c

    how to show that if delta (b square minus 4ac) is a perfect square

    then such a quadratic expression can be factorized by cross method?

    into something like (mx-n)(px-q) where m,n,p,q are integers.

    I had been thinking about it for a whole night.

    Hope someone can help
    If b^2-4ac = d^2 then c = \frac{b^2-d^2}{4a} (note if a = 0, then we don't have a quadratic). So

    ax^2+bx+c = ax^2 + bx + \frac{b^2-d^2}{4a} = \frac{4a^2x^2 + 4abx + b^2 -d^2}{4a}

    From the first three terms in the numerator and then the difference of squares

     <br />
\frac{(2ax +b)^2 -d^2}{4a} = \frac{(2ax + b - d)(2ax + b + d)}{4a}<br />
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