Find all positive values for k for which the following can be factored.x^2 + x - k?

Question

this is like a question I am completely stuck on can someone show me what the heck to do?

Answer 1

I'm not sure what it really means to find all positive values for k, given the fact that the equation says that you have a -k in the term to be factored. The equation can be rewritten as x^2+x+(-k), so you know that you will have no values for k that are positive! However, I will just look at the problem ignoring this fact!

You are looking for two numbers that are multiplied together to equal a -k and when added together equal a +1(coefficient of x)

so let the numbers be a and b
a*b= -k
a+b=1

from above two equations, a&b must be of opposite signs (to get a negative answer) and that a must be 1 greater than b to get a positive 1 when added together

so (+2,-1) k = -2, (+3,-2) k=-6, (+4,-3) k=12 etc.

Hope this is an easier way to see the answer!

Answer 2

Note this: x² + x - k = (x + 1) * (x - k) or (x -1) * (x + k).

In the first case, the middle term is x - kx = 1x. Factoring out the x from the last equation, we get this: (1 - k)x = 1x. That implies that 1 - k = 1. So k = 1 - 1 = 0.

In the second case, the middle term is kx - x = 1x. Again factoring out the x, we get this: (k - 1)x = 1x, which implies that k - 1 = 1. So, in this case, k = 1 + 1 = 2.

So k = 0 or k = 2. To verify that this is correct, substitute these values into the original equation and see if they work.

Let k = 0. Then (x + 1) * (x - 0) = x² + (1 - 0)x - 1(0) = x² + x - 0. So this is true for k = 0.

Now let k = 2. Then (x - 1) * (x + 2) = x² + (2 - 1)x - 1(2) = x² + x - 2. So this is also true for k = 2.

Now we come to a tricky part of the question. It asks for all the positive values of k for which this is true. Is 0 a positive number? No! Zero is neither positive nor negative. So it is not a solution by the restrictions given in the question, even though it does make the equation true. That leaves k = 2 as the only positive value which allows the equation to be factored as given.