Hi! Sorry to bother you all, I was hoping someone could explain the basic rules

Question

Hi! Sorry to bother you all, I was hoping someone could explain the basic rules of Simplifying rational expressions for me as it's been a couple years since I've done this kind of math.

Here's an example:

My main confusion about questions such as these are:

1. In this question, why are they multiplying the top by -2x*3x to get 6x?

2. Why is 9 turning into 3^2?

3.In the denominator, they use addition to get the "-x" and "-7x" terms, why did they not multiply like they did at the top?

Lastley, i'm confused as to how they got all the numbers within the brackets at the denominator. Could someone explain how this was done?

Sorry for the bother, I'm sure this is all very basic stuff!

Given Step 1 of 1 A 3x 6 (Factor) x-6) (2x 7 +3) 3 (x+2) X2 -2.3x +3 6x X 3 (x-3) 1.3(x+2) (Cancellation of like terms) (2x-1)

Explanation / Answer

to simplify rational expression we have to factorise all the mini expression in it

1)formula:x2-2ax+a2=(x-a)2

for x2-6x+9

2a =6 => a=3

and a2=9

so  x2-6x+9 is written as (x-3)2 or (x-3)*(x-3)

2)

for 3x+6, take 3 as common factor

so 3x+6=(3*x)+(3*2)=3*(x+2)

3)comparing with ax2+bx +c.we have to find factors of a*c such that a+c=b

for x2-x-6

factors of 1*-6=-6

-6 can be written as (1*-6),(-1*6),(2*-3),(-2*3)

factors are (1,-6),(-1,6),(2,-3),(-2,3)

for the set (2,-3) we get 2-3 =-1=b

so x2-x-6

=x2+(2-3)x+(2*-3)

=x2+2x-3x+(2*-3)

=(x2+2x)+(-3x+(2*-3))

take out common terms

=(x*(x+2))+(-3*(x+2))

take out common terms

=(x+2)*(x-3)

4)similarly for (2x2-7x+3)

2*3 =6

6 can be written as (2*3) ,(-2*-3),(1*6),(-1*-6)

here -7 =(-1)+(-6)

so (2x2-7x+3)=2x2+(-1-6)x+3

=2x2-1x-6x+3

=(x*(2x-1))+(-3*(2x-1))

=(2x-1)(x-3)

so now

(x2-6x+9)/(x2-x-6)*(3x+6)/(2x2-7x+3)

= [(x-3)*(x-3)/ (x+2)*(x-3)]*[(3*(x+2))/(2x-1)(x-3)]

= [(x-3)/ (x+2)]*[3*(x+2)/(2x-1)(x-3)]

= [(x-3)*3*(x+2)/(x+2)*(2x-1)*(x-3)]

= 3/(2x-1)

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