Question 92349: Find all positive values for k for which each of the following can be factored.
x^2 + x - k
Found 2 solutions by Nate, ptaylor:
Answer by Nate(3500)   (Show Source):
You can put this solution on YOUR website! x^2 + x - k
That means you want factors of -k that sum to 1....
Plug:
~~~ No Negative Values Work
k = -2 ... nope
k = -1 ... nope
k = 0 ... yes
k = 1 ... nope
k = 2 ... yes
k = 3 ... nope
k = 4 ... nope
k = 5 ... nope
k = 6 ... yes
k = 7 ... nope
k = 8 ... nope
k = 9 ... nope
k = 10 ... nope
k = 11 ... nope
k = 12 ... yes
~~~Skip a Few
k = 20 ... yes
Works: 2,6,12,20 ....
+2,+4,+6,+8,+10 ....
Arithmetic Sum:
S(k) = (k/2)(k1 + kx) ... where k1 is the first term and kx is the last term
S(k) = (k/2)(2 + k)
S(k) = k + k^2
First term: S(1) = 1 + 1^2 = 2
Second term: S(2) = 2 + 2^2 = 6
Kth term: S(k) = k + k^2 including zero
Answer by ptaylor(2198)  (Show Source):
You can put this solution on YOUR website! x^2 + x - k----Quadratic in standard form where:
A=1
B=1
C=k
When we have a quadratic where the A coefficient is equal to 1, then the factors of C must add up to the B coefficient. For example, lets assume that the factors for your equation are:
(x+a)(x-b)=0
Expand using the FOIL concept(First, Outer, Inner, Last)
x^2-bx+ax-ab=0 rearrange
x^2+(a-b)x-ab=0--------------eq1
Now in your problem, clearly (a-b) must equal 1
So, if a-b=1 in eq1, then b=(a-1)
Substitute b=(a-1) into eq1 and rewrite it
x^2+(a-(a-1))x-a(a-1)=0 simplify
x^2+(a-a+1)x-a(a-1)=0 or, further simplifying
x^2+x-a(a-1)=0-----------eq2 Now we know that k=a(a-1) where a has to be greater than or equal to zero.
Hope this helps--------------ptaylor
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