Find all positive values for k which each for each of the following can be factored.
(1) x^2 + x + k)
k<= 1/4
To find the values of k for which the quadratic expression x^2 + x + k can be factored, we need to consider the discriminant of the quadratic equation.
The discriminant of a quadratic equation ax^2 + bx + c = 0 is given by the formula: Δ = b^2 - 4ac.
In this case, the quadratic expression is x^2 + x + k, so the coefficients are: a = 1, b = 1, and c = k.
For the quadratic equation to be factorable, the discriminant Δ must be a perfect square. So, Δ = 1 - 4(k).
Since we want to find the positive values of k, we need to ensure that Δ is positive as well. Therefore, we have the inequality Δ > 0.
Substituting the values into the inequality, we get:
1 - 4(k) > 0
Solving for k:
-4k > -1
k < 1/4
So, the positive values for k for which the expression x^2 + x + k can be factored are k < 1/4.