SOLUTION: x^2-5x+6/x^2-4x find the common numerator

Algebra ->  Rational-functions -> SOLUTION: x^2-5x+6/x^2-4x find the common numerator      Log On


   

Question 689925: x^2-5x+6/x^2-4x find the common numerator
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
%28x%5E2-5x%2B6%29%2F%28x%5E2-4x%29
first factor completely numerator %28x%5E2-5x%2B6%29
Solved by pluggable solver: Factoring Quadratics with a leading coefficient of 1 (a=1)
In order to factor 1%2Ax%5E2%2B-5%2Ax%2B6, first we need to ask ourselves: What two numbers multiply to 6 and add to -5? Lets find out by listing all of the possible factors of 6


Factors:

1,2,3,6,

-1,-2,-3,-6,List the negative factors as well. This will allow us to find all possible combinations

These factors pair up to multiply to 6.

1*6=6

2*3=6

(-1)*(-6)=6

(-2)*(-3)=6

note: remember two negative numbers multiplied together make a positive number

Now which of these pairs add to -5? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -5

||||
First Number|Second Number|Sum
1|6|1+6=7
2|3|2+3=5
-1|-6|-1+(-6)=-7
-2|-3|-2+(-3)=-5
We can see from the table that -2 and -3 add to -5.So the two numbers that multiply to 6 and add to -5 are: -2 and -3 Now we substitute these numbers into a and b of the general equation of a product of linear factors which is: %28x%2Ba%29%28x%2Bb%29substitute a=-2 and b=-3 So the equation becomes: (x-2)(x-3) Notice that if we foil (x-2)(x-3) we get the quadratic 1%2Ax%5E2%2B-5%2Ax%2B6 again


and we can factor denominator and write it as x%28x-4%29

now we have %28%28x-2%29%28x-3%29%29%2Fx%28x-4%29...this is all we can do and you can see there are no common factors in nominator and denominator to be canceled, and this is the simplest way to write your expression