SOLUTION: This question is on Rational Expressions, and solving them using the Algebraic method to find the vertical asymptote. g(x)=(x^2+x+1)/(5x^2+1) I can't figure out where to start.

Algebra ->  Rational-functions -> SOLUTION: This question is on Rational Expressions, and solving them using the Algebraic method to find the vertical asymptote. g(x)=(x^2+x+1)/(5x^2+1) I can't figure out where to start.      Log On


   

Question 185346: This question is on Rational Expressions, and solving them using the Algebraic method to find the vertical asymptote.
g(x)=(x^2+x+1)/(5x^2+1) I can't figure out where to start...
Also simplifying (3a^2-6a)/(12-6a) What the heck is this? And where do I start and finish this crazy problem???
Found 2 solutions by vleith, jim_thompson5910:
Answer by vleith(2983) About Me  (Show Source):
You can put this solution on YOUR website!
Vertical asymptotes occur as the value of denominator approaches 0 (which causes the quotient to approach infinity).
So find the values that make the denominator 0
5x%5E2+%2B1=0
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 5x%5E2%2B0x%2B1+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%280%29%5E2-4%2A5%2A1=-20.

The discriminant -20 is less than zero. That means that there are no solutions among real numbers.

If you are a student of advanced school algebra and are aware about imaginary numbers, read on.


In the field of imaginary numbers, the square root of -20 is + or - sqrt%28+20%29+=+4.47213595499958.

The solution is

Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+5%2Ax%5E2%2B0%2Ax%2B1+%29

no real values of x cause that to happen, so the answer is 'there are no vertical asymtotes' for this expression.
See this URL for more info --> http://www.purplemath.com/modules/asymtote.htm
Simplify %283a%5E2-6a%29%2F%2812-6a%29
%283a%2A%28a-2%29%29%2F%28%28-6%29%2A%28a-2%29%29
-a%2F2

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
# 1


g%28x%29=%28x%5E2%2Bx%2B1%29%2F%285x%5E2%2B1%29 Start with the given function.


5x%5E2%2B1=0 Set the denominator equal to zero. Remember, vertical asymptotes arise only when there's a division by zero


5x%5E2=-1 Subtract 1 from both sides.


x%5E2=-1%2F5 Divide both sides by 5


x=sqrt%28-1%2F5%29 Take the square root of both sides.


Remember, you CANNOT take the square root of a negative number. So this means that the value of "x" will be complex/imaginary.


In other words, there is NO value of "x" that will make the denominator zero.


So there are no vertical asymptotes (graph the function to verify)


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# 2

%283a%5E2-6a%29%2F%2812-6a%29 Start with the given expression


%283a%28a-2%29%29%2F%2812-6a%29 Factor the numerator


%283a%28a-2%29%29%2F%28-6%28a-2%29%29 Factor the denominator


-%28a%28a-2%29%29%2F%282%28a-2%29%29 Reduce 3%2F%28-6%29 (the outer coefficients) to get -1%2F2


-%28a%2Ahighlight%28%28a-2%29%29%29%2F%282%2Ahighlight%28%28a-2%29%29%29 Highlight the common terms.


-%28a%2Across%28%28a-2%29%29%29%2F%282%2Across%28%28a-2%29%29%29 Cancel out the common terms.


-a%2F2 Simplify


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Answer:

So %283a%5E2-6a%29%2F%2812-6a%29 simplifies to -a%2F2


In other words, %283a%5E2-6a%29%2F%2812-6a%29=-a%2F2 where a%3C%3E2