can someone show me the steps to work out this problem that is diving me nuts..
Directions: Find all positive values for k for which each of the following can be factored.
x^2+x-k
I am completely cluelis towards this problem....i've tried it many different ways....
x= -1/2 +- sqrt(1+4k) /2
Now if you are restricting the x to the real number system, then k can be any postive number.
The factors will be
(x+1/2-1/2 sqrt(1+4k) ) and (x+1/2
+1/2 sqrt(1+4k)) for any value of k>0
can you explain how to get there because i still don't understand
To factor the quadratic expression, x^2 + x - k, we need to find the values of k for which the expression can be written as the product of two binomials.
Step 1: Identify the factors of k.
Since the coefficient of x^2 is 1, the factors of k must be of the form k = p * q, where p and q are positive integers.
Step 2: Write down the expression with the factors of k.
x^2 + x - k = x^2 + x - p * q
Step 3: Determine the values of p and q.
For the expression to be factorable, we need to find values of p and q such that when multiplied, they give the last term (-k) and when added, they give the coefficient of the middle term (1).
Step 4: List the possible factor pairs.
To find the values of p and q, consider all the possible factor pairs of the last term (-k) and check if their sum is equal to the coefficient of the middle term (1).
For example, if k = 4, the possible factor pairs are (1, 4) and (-1, -4). Checking their sums, we see that (1 + 4) = 5 ≠ 1, and (-1 - 4) = -5 ≠ 1, so neither of these factor pairs work.
Step 5: Find the solution.
Continue checking all the possible factor pairs until you find a pair whose sum is equal to the coefficient of the middle term (1). The values of p and q from this factor pair will give you the factors of the expression.
For example, let's consider k = 5. The factor pairs of 5 are (1, 5) and (-1, -5). Checking their sums, both (1 + 5) = 6 = 1 and (-1 - 5) = -6 = 1 are equal to the coefficient of the middle term (1). So, p = 1 and q = 5 will give us the factors of the expression.
Substituting the values of p and q back into the expression, we get:
x^2 + x - k = x^2 + x - 1 * 5 = x^2 + x - 5
The factors of x^2 + x - 5 are (x + 2)(x - 5).
So, for k = 5, the expression x^2 + x - k can be factored as (x + 2)(x - 5).
Repeat this process for different values of k to find all positive values for which the expression can be factored. In the end, you will find that for any positive value of k, the expression x^2 + x - k can be factored as (x + 1/2 - 1/2 * sqrt(1 + 4k))(x + 1/2 + 1/2 * sqrt(1 + 4k)).