# Simplify ( 2x^2 - 8 )/( x - 2 ) + 13/x( x -2 ) - ( 4x^2 - 16 )/( x + 2 )

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In order to calculate the sum or difference of 3 ratios, we have to verify if they have a common denominator.

But, before verifying if they have a common denominator, we'll solve the difference of squares from the brackets.

We'll factorize by 2 the first ratio:

( 2x^2 - 8 )/( x - 2 ) = 2(x^2 - 4)/( x - 2 )

We'll write the difference of squares (x^2 - 4) as a product:

(x^2 - 4) = (x-2)(x+2)

We'll re-write the ratio;

( 2x^2 - 8 )/( x - 2 ) = 2(x-2)(x+2)/(x - 2)

We'll reduce like terms:

( 2x^2 - 8 )/( x - 2 ) = 2(x+2)

We'll factorize by 4 the third ratio:

( 4x^2 - 16 )/( x + 2 ) = 4(x^2 - 4)/( x + 2 )

4(x^2 - 4)/( x + 2 ) = 4(x-2)(x+2)/(x + 2)

We'll reduce like terms:

4(x^2 - 4)/( x + 2 ) = 4(x-2)

We'll re-write now the given expression:

2(x+2) + 13/x( x -2 ) - 4(x-2)

It's obvious that the least common denominator (LCD) is the denominator of the second ratio:

To calculate the expression we'll do the steps:

- we'll multiply 2(x+2) by x*(x-2)

- we'll multiply 4(x-2) by x*(x-2)

We'll get:

2(x+2)*x*(x-2) + 13 - 4(x-2)*x*(x-2)

We'll open the brackets:

2x^3 - 8x + 13 - 4x^3 + 16x^2 - 16x

We'll group like terms and we'll get:

**-2x^3 + 16x^2 - 24x + 13**

Here we use the result that x^2-y^2=(x-y)(x+y) extensively.

(2x^2-8)/(x-2)+13/x(x-2)-(4x^2-16)/(x+2)

=2(x-2)(x+2)/(x-2)+13/x(x-2)- 4(x-2)(x+2)/(x+2)

=2(x+2)+13/x(x-2)-4(x-2)

=[2x(x+2)(x-2)+13-4x(x-2)] / [x(x-2)]

=[2x(x^2-4)+13-4x^2+8x]/ [x(x-2)]

=[2x^3-8x+13-4x^2+8x]/ [x (x-2)]

(2x^3-4x^2+13)/[x(x-2)]

**The simplified result is (2x^3-4x^2+13) / [x(x-2)]**

To Simplify :

( 2x^2 - 8 )/( x - 2 ) + 13/x( x -2 ) - ( 4x^2 - 16 )/( x + 2

2(x^2-8)/(x+2) = 2(x^2-2^2)/(x-2) = 2(x+2)(x-2)/(x-2) = 2(x+2)

(4x^2-16)/(x+2) =3(x^2-2^2)/(x+2) = 4(x+2)(x-2)/(x+2) = 4(x-2).

Therefore the given expression becomes:

2(x+2) +13/(x(x-2) +4(x-2)

= 4x+4 +4x-8 +13/x(x-2)

= (8x-4) +13/x(x-2)

={8x-4)x(x-2)+13}/(x(x-2)

= {8x^3 -20x+8x+13}/(x(x-2))