How do I find a value for k that will make the expression factorable.
36x^2 + 8x + k?
Clearly, k=0 will do the job, but I doubt that's what you want.
the quadratic formula says
x = (-8±√(64-144k))/72
So, any negative value of k which makes 64-144k a perfect square will work. SO, if we let v be 64-144k, we have
k v
-1 208
-2 352
-3 496
-4 640
-5 784 = 28^2
36x^2+8x-5 = (2x+1)(18x-5)
There are, of course, many other workable values.
To find a value for k that will make the expression factorable, first we need to understand the concept of factorability. An expression is considered factorable if it can be written as a product of two binomial factors. In other words, we are looking for values of k that will allow us to rewrite the expression in the form "(ax + b)(cx + d)".
To determine these values, we can use a method called factoring by grouping. Here's a step-by-step process to find the value for k:
1. Start with the given expression: 36x^2 + 8x + k.
2. We need to factor out the common term from the first two terms, which is 4x: 4x(9x + 2) + k.
3. Now, let's take a look at the expression (9x + 2). To make it factorable, we need to find two numbers that multiply to give k and add up to give us the middle coefficient (8x). Let's call these numbers a and b.
4. We can set up the equation: a * b = k and a + b = 8.
5. One way to approach this is to try different values of a and b until we find a combination that satisfies both conditions. You can start by trying small values for a and b, such as a = 1 and b = 7. This would give us k = 1 * 7 = 7 and 1 + 7 = 8, which satisfies both conditions.
6. Once you have found the values for a and b, substitute them back into the expression to factor it: 4x(9x + 2) + k = 4x(9x + 2) + 7.
So, the value of k that will make the expression 36x^2 + 8x + k factorable is k = 7.