x^2+x-k
find all positive values for k, if it can be factored
Allowed values of k for for the expression to be factorable with integers are
a*(a+1)
where a is an integer equal to 1 or more.
For example: 2, 6, 12, 20...
The factors are
(x-a)(x+a+1)
There are also real non-integer values of k that allow factrs, as long as 1+4k >0
To determine the allowed values of k for the expression to be factorable, we need to find the quadratic factors of the expression x^2 + x - k.
First, let's rewrite the quadratic expression in the form of (x - a)(x + b), where a and b are integers.
Expanding this form, we have:
x^2 + x - k = (x - a)(x + b)
To find a and b, we can compare the coefficients of the original expression and the expanded form.
Comparing the x^2 terms:
1 (coefficient of x^2) = 1 (coefficient of x in (x - a)(x + b))
Comparing the x terms:
1 (coefficient of x) = -a + b (sum of coefficients of x)
Comparing the constant terms:
- k (constant term in the original expression) = -a * b (product of the constant terms)
From the first comparison, we see that the coefficient of x^2 is the same in both expressions. This tells us that a must be equal to 1.
Substituting a = 1 into the second equation, we get:
1 = -1 + b
b = 2
So, the quadratic expression x^2 + x - k can be factored as:
(x - 1)(x + 2)
To find the allowed values of k, we need to consider integers a and b. In this case, a = 1 and b = 2.
The allowed values of k are given by the product of a and b:
k = 1 * 2 = 2
Therefore, the only positive value for k that allows the expression x^2 + x - k to be factored with integers is k = 2.
Note: In addition to the above, there may also be real non-integer values of k that allow factoring, as long as 1 + 4k > 0.