SOLUTION: please help me with this: For a value of K, the expression x+k/x^2+kx-15 is undefined when x=5 and x=-3. Find the value of K

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: please help me with this: For a value of K, the expression x+k/x^2+kx-15 is undefined when x=5 and x=-3. Find the value of K      Log On


   

Question 6601: please help me with this:

For a value of K, the expression x+k/x^2+kx-15 is undefined when x=5 and x=-3. Find the value of K
Answer by prince_abubu(198) About Me  (Show Source):
You can put this solution on YOUR website!
The trick here lies in the denominator. Hang on for the ride. This is a bit tricky.

I take it that you know that 5 and -3 will make the expression fail because those two values will force the denominator to be equal to 0.

They already gave you the fact that the denominator is quadratic (because of the x^2 term), and that quadratic expression can be factored in (x + _)(x + _) form (reverse-FOILING). We now need to set that denominator equal to zero to find which values of x make that denominator zero. In other words, we are looking for illegal values of x. So far, we have the equation

+%28x+%2B+_%29%28x+%2B+_%29+=+0+

But what values do you put in place of the blanks? Ah! They gave you the x-values already - the 5 and the -3. So, substitute the 5 and the -3 for each of the x's, NOT the blanks. Now we have

+%285+%2B+_%29%28-3+%2B+_%29+=+0+

Remember that 0 times anything equals zero. So we'll pick at the first set of parentheses and set it equal to zero:

+%285+%2B+_%29+=+0+ You'll see that _ = -5

Let's take the second set of parentheses and set it equal to zero.

+%28-3+%2B+_%29+=+0+ You'll see that _ = 3

Now rewrite the equation with the x's and place the two values you found for the blanks:

+%28x+-+5%29%28x+%2B+3%29+=+0+

NOTE: We could've just agreed to write write the quadratic in (x + _)(x + _) form and just plug in the undefined values given and change the signs. So we could've straight up said (x - 5)(x + 3) = 0 (plug in the numbers but reverse the signs just because it works out that way), but I just wanted to show you why and how it works that way.

Now, do FOIL on the (x - 5)(x + 3). This turns out to be +x%5E2+-+2x+-+15+. Now put that head to head with +x%5E2+%2B+kx+-+15+ and you'll see that k = -2.

WARNING. Don't let the "+ k" vs. the "- 2" fool you into believing that there is no way to work out this problem. When you see "+ k" or "+ any variable", it just means that k can be any number. If k turned out to be negative, like in this case, it really is "+ -2" which turns out to be "- 2".