Find all positive values for k for which each of the following can be factored.
X^2-x-k
(Your answer seems to be the same as the other problem x^2+x-k )
Is this correct too?
Consider your equation to be in the form ax^2 + bx = c = 0. It can be factored (that is, have real answers for x) if the term b^2 - 4ac is positive or zero. In your case,
b^2 - 4ac = 1 + 4k, so 1+4k >or= 0
k >or= -1/4
If k = -1/4, you have the special case
(x-1/2)^2 = 0, with only one root, x = 1/2
To find the values of k for which the equation x^2 - x - k can be factored, we can use the quadratic formula.
First, let's rearrange the equation to match the standard quadratic form ax^2 + bx + c = 0.
x^2 - x - k = 0
In this case, a = 1, b = -1, and c = -k. Now, we can plug these values into the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Applying it to our equation, we get:
x = (1 ± √((-1)^2 - 4(1)(-k))) / (2(1))
x = (1 ± √(1 + 4k)) / 2
For this equation to have real solutions, the term under the square root, 1 + 4k, must be greater than or equal to zero.
1 + 4k ≥ 0
To solve this inequality, we subtract 1 from both sides:
4k ≥ -1
Then divide by 4:
k ≥ -1/4
So, the positive values of k for which the equation x^2 - x - k can be factored are k ≥ -1/4.
Regarding the equation x^2 + x - k, the process is the same.
x^2 + x - k = 0
Again, a = 1, b = 1, and c = -k.
Using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
We get:
x = (-1 ± √(1^2 - 4(1)(-k))) / (2(1))
x = (-1 ± √(1 + 4k)) / 2
The term under the square root, 1 + 4k, must be greater than or equal to zero for real solutions.
1 + 4k ≥ 0
Following the same steps as before, we find:
k ≥ -1/4
So, the positive values of k for which the equation x^2 + x - k can be factored are also k ≥ -1/4.
Therefore, both equations have the same positive values for k, which are k ≥ -1/4.