SOLUTION: Find all positie values for K for which each of the following can be factored. x^2 + x - k Thank you for your help.

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Question 27864: Find all positie values for K for which each of the following can be factored.
x^2 + x - k
Thank you for your help.
Answer by bmauger(101) About Me  (Show Source):
You can put this solution on YOUR website!
It can only be factored if the discriminant is positive and a perfect square. The discriminant is defined by:
b%5E2-4ac for ax%5E2%2Bbx%2Bc
For your problem x%5E2+%2B+x+-+k a=1, b=1 and c=-k
so the descriminant is 1%2B4k
So any number that when multiplied by 4 is one less than a square number would work. Such as:
0 --> 1+4*0=1 --> x%5E2%2Bx=%28x%2B1%29%28x%29
2 --> 1+4*2=9 --> x%5E2%2Bx-2=%28x%2B2%29%28x-1%29
6 --> 1+4*6=25 --> x%5E2%2Bx-6=%28x%2B3%29%28x-2%29
12 --> 1+4*12=49 --> x%5E2%2Bx-12=%28x%2B4%29%28x-3%29
20 --> 1+4*20=81 --> x%5E2%2Bx-20=%28x%2B5%29%28x-4%29
30 --> 1+4*30=121 --> x%5E2%2Bx-30=%28x%2B6%29%28x-5%29
42 --> 1+4*42=169 --> x%5E2%2Bx-42=%28x%2B7%29%28x-6%29
Starting to see some paterns, no doubt?...
So there's an infinite number of k's that'll work. To find the next value in the pattern, square each odd number, subtract one, and divide by 4.
%281%5E2-1%29%2F4=0
%283%5E2-1%29%2F4=2
%285%5E2-1%29%2F4=6
%287%5E2-1%29%2F4=12
%289%5E2-1%29%2F4=20
%2811%5E2-1%29%2F4=30
%2813%5E2-1%29%2F4=42
%2815%5E2-1%29%2F4=56
%2817%5E2-1%29%2F4=72
Answer your question?