SOLUTION: I have learned how to completely factor polynomials such as: x^2+7x-30 (x+10)(x-3) but I get lost when there are variables instead of number. I need to find all the positiv

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: I have learned how to completely factor polynomials such as: x^2+7x-30 (x+10)(x-3) but I get lost when there are variables instead of number. I need to find all the positiv      Log On


   

Question 73374: I have learned how to completely factor polynomials such as:
x^2+7x-30
(x+10)(x-3)
but I get lost when there are variables instead of number. I need to find all the positive values for "k" in this problem:
x^2+x-k
Answer by psbhowmick(878) About Me  (Show Source):
You can put this solution on YOUR website!
x%5E2%2Bx-k
= x%5E2%2B2x%281%2F2%29%2B%281%2F2%29%5E2-%281%2F2%29%5E2-k
= %28x%2B1%2F2%29%5E2-%281%2F2%29%5E2-k
= %28x%2B1%2F2%29%5E2-1%2F4-k
= %28x%2B1%2F2%29%5E2-%281%2F4%2Bk%29

This can be factorised if k%2B1%2F4 is a perfect square.
Thus, those positive values of k which render k%2B1%2F4 a perfect square are the reqd. values of k.
So, k = 2, 6, 12, 20, 30, etc.

[Note that the difference between the consecutive terms form an A.P.: 4,6,8,10,.... ]