How do you simplify #(x-2)/(5x(x-1)) + 1/(x-1) - (3x+2)/(x^2+4x-5)#?

Question

Avery Alexander · Accepted Answer

#= (15 - 9x^2 - 7x)/(5x^3 + 20x^2 - 25x)#

#color(white)(xx)(x - 2)/(5x(x -1)) + 1/(x - 1) - (3x + 2)/(x^2 + 4x - 5)#

#= (x - 2)/(5x(x -1)) + 1/(x - 1) - (3x + 2)/(x^2 - x +5x - 5)#

[Broken up #4x# as #-x + 5x#]

#=(x - 2)/(5x(x -1)) + 1/(x - 1) - (3x + 2)/(x(x - 1) + 5(x - 1))#

#=(x - 2)/(5x(x -1)) + 1/(x - 1) - (3x + 2)/((x-1)(x+5))#

#=((x - 2)(x + 5) + 5x(x + 5) -5x(3x + 2))/(5x(x-1)(x+5))# [Taking The LCM]

#=((x^2 - 2x + 5x - 10) + (5x^2 + 25) -(15x^2 + 10x))/(5x(x-1)(x+5))#

#= (x^2 - 2x + 5x - 10 + 5x^2 + 25 -15x^2 - 10x)/(5x(x-1)(x+5))#

#= (-9x^2 - 7x + 15)/(5x^3 + 20x^2 - 25x)#

#= (15 - 9x^2 - 7x)/(5x^3 + 20x^2 - 25x)#

Hope this helps.

Jameson Allen · Answer

undefined To simplify the expression (x-2)/(5x(x-1)) + 1/(x-1) - (3x+2)/(x^2+4x-5), we need to find a common denominator and combine the fractions.

The common denominator for the three fractions is 5x(x-1)(x+5)(x-1).

Multiplying the first fraction by (x+5)/(x+5), the second fraction by 5x(x+5)/(5x(x+5)), and the third fraction by 5x(x-1)(x+5)/(5x(x-1)(x+5)), we get:

[(x-2)(x+5)]/[5x(x-1)(x+5)] + [5x(x+5)]/[5x(x-1)(x+5)] - [(3x+2)(5x(x+5))]/[5x(x-1)(x+5)]

Expanding and simplifying each fraction, we have:

[(x^2+3x-10)]/[5x(x-1)(x+5)] + [5x^2+25x]/[5x(x-1)(x+5)] - [15x^2+25x]/[5x(x-1)(x+5)]

Combining the fractions, we get:

[(x^2+3x-10) + (5x^2+25x) - (15x^2+25x)]/[5x(x-1)(x+5)]

Simplifying the numerator, we have:

(x^2+3x-10 + 5x^2+25x - 15x^2-25x)/[5x(x-1)(x+5)]

Combining like terms, we get:

(-9x^2+3x-10)/[5x(x-1)(x+5)]

Therefore, the simplified expression is (-9x^2+3x-10)/[5x(x-1)(x+5)].