sjuniv32 wrote:
If \(c\) is a constant, and \(a\) and \(b\) are solutions for \(x\) to the equation \(x^2 + 2cx + c^2 = 1\), then which of the following could be equal to \(a − b\)?
I) 1
II) 2
III) 4
A) I only
B) II only
C) III only
D) I and III only
E) II and III only
Rewrite the equation in standard form, \(x^2 + 2cx + c^2 - 1 = 0\).
Note from the constant part, we have \(c^2 - 1 = (c - 1)(c + 1)\). Then also note if we add up \(c - 1\) and \(c + 1\) we get exactly \(2c\), the coefficeint of the x term. So we can actually factor this equation into \((x - c + 1)(x - c - 1) = 0\). Thus the solutions in terms of c would be \(x = c - 1\) and \(x = c + 1\).
Finally, if we subtract the two solutions from each other we can only get 2 or -2.
Ans: B
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