Find all positive values for k for which each of the following can be factored.
X^2-x-k
Using the quadratic equation,
x= 1/2 +-1/2* sqrt (1+4k)
Now, the sqrt has to be of a positive number, so 1+4k>0 so any value of
k>-1/4 should work.
To find the positive values of k for which the expression X^2 - x - k can be factored, we need to determine the conditions under which the quadratic equation x = 1/2 ± 1/2√(1 + 4k) remains valid.
Using the quadratic formula, x = (-b ± √(b^2 - 4ac))/2a, where the equation is in the form ax^2 + bx + c = 0, we can equate our expression to zero:
X^2 - x - k = 0
Comparing this to the quadratic formula format, we have a = 1, b = -1, and c = -k. Plugging these values into the quadratic formula, we get:
x = (1 ± √((-1)^2 - 4(1)(-k)))/2(1)
x = (1 ± √(1 + 4k))/2
Now, for x to be a real number, the expression inside the square root (√(1 + 4k)) must be non-negative. In other words, 1 + 4k ≥ 0. Solving this inequality for k, we have:
4k ≥ -1
k ≥ -1/4
Therefore, any value of k greater than or equal to -1/4 should satisfy the condition for factoring the expression X^2 - x - k. Since we are looking for positive values of k, the final answer is k > -1/4.