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Math Help - factorising x^2 - x + 4

  1. #1
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    factorising x^2 - x + 4

    Hi,

    I have been trying to factorise the above, but simply cannot find a solution.

    Thanks for any help.
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: factorising x^2 - x + 4

    Quote Originally Posted by bermudj View Post
    I have been trying to factorise the above, but simply cannot find a solution.
    The roots z_1,z_2 of p(x)=x^2-x+4 are not real so, p(x) is irreducible on \mathbb{R} . On \mathbb{C} its factorization is p(x)=(x-z_1)(x-z_2) .
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  3. #3
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    Re: factorising x^2 - x + 4

    Quote Originally Posted by bermudj View Post
    Hi,

    I have been trying to factorise the above, but simply cannot find a solution.

    Thanks for any help.
    \displaystyle \begin{align*} x^2 - x + 4 &= x^2 - x + \left(-\frac{1}{2}\right)^2 - \left(-\frac{1}{2}\right)^2 + 4 \\ &= \left(x - \frac{1}{2}\right)^2 - \frac{1}{4} + \frac{16}{4} \\ &= \left(x - \frac{1}{2}\right)^2 + \frac{15}{4} \\ &= \left(x - \frac{1}{2}\right)^2 - \left(\frac{i\sqrt{15}}{2}\right)^2 \\ &= \left(x - \frac{1}{2} - \frac{i\sqrt{15}}{2}\right)\left(x - \frac{1}{2} + \frac{i\sqrt{15}}{2}\right) \end{align*}
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