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I'm reading Elementary Algebra by Rouse W. W. Ball, and factorization strategies for quadratics are being discussed.

In the book, we are trying to factor $x^2+2x-1$, so we complete the square

$$x^2+2x-1$$ $$=(x^2+2x+1)-1-1$$ $$=(x+1)^2-2$$

But the author doesn't stop there; he says that we can use difference of squares to factor this further.

$$a^2-b^2=(a+b)(a-b)$$ And thus our equation becomes $$(x+1-\sqrt2)(x+1+\sqrt2)$$

But what bothers me is that the author then says: "These factors are rational so far as $x$ is concerned." I have no idea what this means.

Aren't these factors irrational since $\sqrt2$ is involved?

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  • $\begingroup$ What does the author say after that? $\endgroup$
    – Raad Shaikh
    Mar 25 at 19:47
  • $\begingroup$ He moves on to how to complete the square when the coefficient of $x^2$ is not 1, i.e., on quadratics of the form $ax^2+bx+c$ (not $x^2+bx+c$) $\endgroup$
    – UserM1
    Mar 25 at 19:52
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    $\begingroup$ My best guess is that he's trying to say there's nothing irrational attached to $x$ as a coefficient. Although admittedly I'm not sure why that would be important ^_^ $\endgroup$
    – Raad Shaikh
    Mar 25 at 19:56
  • $\begingroup$ I agree with you. I don't know why that would be important either. $\endgroup$
    – UserM1
    Mar 25 at 20:02
  • $\begingroup$ Perhaps „it is a product of polynomials in $x$“ as opposed to „it involves square roots of polynomials in $x$“? $\endgroup$
    – Hagen von Eitzen
    Mar 26 at 10:42

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