Posted by Math 10 on Saturday, October 31, 2009 at 10:04pm.
It can be factored, but not into integers like (x-1); more like sqrt(2)+-1, and it's definitely not a perfect square. I suspect you've got one of the signs backwards in your head, and you'll probably recognise that as we work through it.
Anyway, direct factorisation wasn't the question; this one is about completing the square.
Forget the constant for a minute, and focus on x^2-2x.
x^2-2x+c is a square. We just have to find c.
We know that
(x+a)^2 = x^2 + 2ax + a^2
and we want
x^2 - 2x + c
so, comparing the coefficients:
2a = -2
a = -1
(x-1)^2 = x^2 - 2x + 1
Our expression is 2 less than that, so:
x^2-2x-1 = (x-1)^2 - 2
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