I think I just need some background.
I've got the following quadratic equation:
$$ 1 - x - 2x^2 = (1-2x)(1 + x) $$
But if I solve it with the quadratic equation, I get the roots:
$$ \frac{1 + \sqrt{9}}{-4} = -1, \frac{1-\sqrt{9}}{-4} = +1/2 $$
And, logically, I can't think of why I wouldn't write:
$$ (1 + x)(1/2 - x) = \frac{1}{2} - \frac{1}{2}x - x^2 = \frac{1}{2}(1 - x - 2x^2) $$
So I guess my question is:
What is the standard I should be holding to when I recreate the function from the roots, so that this kind of mistake doesn't happen? (do I always start with $(1-ax)$ and solve for $a$ when $x = \text{the root}$, for example?)
And...does it matter? Obviously, the second equation just looks different...the relationships are the same...but I'd like to be operating the 'regular' way...
If $\;\alpha,\,\beta\;$ are the roots of the quadratic $\;y=ax^2+bx+c\;$ , then you get
$$ax^2+bx+c=a(x-\alpha)(x-\beta)$$
I think you just forgot the higher coefficient $\;a\;$ . You may want to google Vieta's Formulas