Question

With full explanation from the internet like before
1/(x-5)+3/(x+2)=4

Answer 1

`\\ \rm\Rrightarrow (1)/(x-5)+(3)/(x+2)=4`

`\\ \rm\Rrightarrow (x+2+3x-15)/((x-5)(x+2))=4`

`\\ \rm\Rrightarrow 4x-13=4(x-5)(x+2)`

`\\ \rm\Rrightarrow 4x-13=4(x(x+2)-5(x+2))`

`\\ \rm\Rrightarrow 4x-13=4(x^2+2x-5x-10)`

`\\ \rm\Rrightarrow 4x-13=4(x^2-3x-10)`

`\\ \rm\Rrightarrow 4x-13=4x^2-12x-40`

`\\ \rm\Rrightarrow 4x^2-16x-53=0`

On solving we get

`\\ \rm\Rrightarrow x=2\pm(69)/(2)`

Answer 2

`x = (4 + √(43) )/(2) .x = 4 + (4 - √(43) )/(2)`

Explanation:

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

1. Multiply by LCM

`x = 2 + 3( x - 5) = 4 - (x - 5)(x + 2)`

2. Solve

`x = 2 + 3(x - 5) = 4 (x - 5)(x + 2)`

`x = (4 + √(43) )/(2) .x = 4 + (4 - √(43) )/(2)`

3.Verify Solutions

Find undefined (singularity) points :

x=5,x=–2

`x = (4 + √(43) )/(2) .x = 4 + (4 - √(43) )/(2)`