Simplify the expression ( x^2 - 4 ) / (5( 2x - 4 )( x -3 )) and leave it in the form ( ax + b ) / ( cx + d ) where a, b, c and d are integers.
Let us start by factorising the numerator and then the denominator separately.Our numerator is x2-4. It is quite obvious that this is the difference of 2 squares. In this case, we can write it as (x-2)(x+2). We can not simplify this any further.
Our denominator is 5(2x-4)(x-3). We can not see anything in common between the three terms, but if we look at 2x-4, we can see that 2 is common between 2x and -4. We can simplify this to 2(x-2) which means our denominator can be written as 5*2(x-2)(x-3) which we can simplify to 10(x-2)(x-3) by multiplying the 5 and 2. Finally we can put the numerator and denominator together and we can clearly see that they both have a x-2 term. So we can go ahead and cancel these two out. This leaves us with (x+2)/(10(x-3)).We can simplify this to the required form as (x+2) / (10x - 30) where a=1 b=2 c=10 and d=-30.
