3x/(4x-1) + 2x/(3x-5)

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We'll multiply the first fraction by (3x - 5) and the 2nd by (4x-1)

[3x(3x - 5) + 2x(4x-1)]/(4x-1)(3x-5)

We'll remove the brackets:

(9x^2 - 15x + 8x^2 - 2x)/(12x^2 - 20x - 3x + 5)

(17x^2 - 17x)/(12x^2 - 23x + 5)

We'l find the roots of the denominator;

12x^2 - 23x + 5 = 0

x1 = (23+sqrt289)/24

x1 = (23+17)/24

x1 = 5/3

x2 = 1/4

17x(x-1)/(x - 5/3)(x-1/4)

Multiply initial fraction (3x - 5) and the 2nd by (4x-1)

[3x(3x - 5) + 2x(4x-1)]/(4x-1)(3x-5)

Remove brackets:

(9x^2 - 15x + 8x^2 - 2x)/(12x^2 - 20x - 3x + 5)

(17x^2 - 17x)/(12x^2 - 23x + 5)

Square the denomiter;

12x^2 - 23x + 5 = 0

x1 = (23+sqrt289)/24

x1 = (23+17)/24

x1 = 5/3

x2 = 1/4

17x(x-1)/(x - 5/3)(x-1/4)

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