SOLUTION: Hello can someone show me how to find the domain of the rational function f(x) = x^2 + 3x -40 / x^2 - 10x +24

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Question 90414: Hello can someone show me how to find the domain of the rational function
f(x) = x^2 + 3x -40 / x^2 - 10x +24
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Set the denominator equal to zero:

x%5E2+-+10x+%2B24=0

Let's use the quadratic formula to solve for x:


Starting with the general quadratic

ax%5E2%2Bbx%2Bc=0

the general solution using the quadratic equation is:

x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29

So lets solve x%5E2-10%2Ax%2B24=0 ( notice a=1, b=-10, and c=24)

x+=+%28--10+%2B-+sqrt%28+%28-10%29%5E2-4%2A1%2A24+%29%29%2F%282%2A1%29 Plug in a=1, b=-10, and c=24



x+=+%2810+%2B-+sqrt%28+%28-10%29%5E2-4%2A1%2A24+%29%29%2F%282%2A1%29 Negate -10 to get 10



x+=+%2810+%2B-+sqrt%28+100-4%2A1%2A24+%29%29%2F%282%2A1%29 Square -10 to get 100 (note: remember when you square -10, you must square the negative as well. This is because %28-10%29%5E2=-10%2A-10=100.)



x+=+%2810+%2B-+sqrt%28+100%2B-96+%29%29%2F%282%2A1%29 Multiply -4%2A24%2A1 to get -96



x+=+%2810+%2B-+sqrt%28+4+%29%29%2F%282%2A1%29 Combine like terms in the radicand (everything under the square root)



x+=+%2810+%2B-+2%29%2F%282%2A1%29 Simplify the square root (note: If you need help with simplifying the square root, check out this solver)



x+=+%2810+%2B-+2%29%2F2 Multiply 2 and 1 to get 2

So now the expression breaks down into two parts

x+=+%2810+%2B+2%29%2F2 or x+=+%2810+-+2%29%2F2

Lets look at the first part:

x=%2810+%2B+2%29%2F2

x=12%2F2 Add the terms in the numerator
x=6 Divide

So one answer is
x=6



Now lets look at the second part:

x=%2810+-+2%29%2F2

x=8%2F2 Subtract the terms in the numerator
x=4 Divide

So another answer is
x=4

So our solutions are:
x=6 or x=4


This since x=6 and x=4 both make the denominator equal to zero, this means that x=6 and x=4 are restricted from the domain. So the domain is the element of all real numbers, excluding x=6 and x=4.