1/(x-3)-4/x-3-5=0

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Solution for 1/(x-3)-4/x-3-5=0 equation:



1/(x-3)-4/x-3-5=0
Domain of the equation: (x-3)!=0
We move all terms containing x to the left, all other terms to the right
x!=3
x∈R
Domain of the equation: x!=0
x∈R
We add all the numbers together, and all the variables
1/(x-3)-4/x-8=0
We calculate fractions
x/(x^2-3x)+(-4x+12)/(x^2-3x)-8=0
We multiply all the terms by the denominator
x+(-4x+12)-8*(x^2-3x)=0
We multiply parentheses
-8x^2+x+(-4x+12)+24x=0
We get rid of parentheses
-8x^2+x-4x+24x+12=0
We add all the numbers together, and all the variables
-8x^2+21x+12=0
a = -8; b = 21; c = +12;
Δ = b2-4ac
Δ = 212-4·(-8)·12
Δ = 825
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:
$\sqrt{\Delta}=\sqrt{825}=\sqrt{25*33}=\sqrt{25}*\sqrt{33}=5\sqrt{33}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(21)-5\sqrt{33}}{2*-8}=\frac{-21-5\sqrt{33}}{-16} $
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(21)+5\sqrt{33}}{2*-8}=\frac{-21+5\sqrt{33}}{-16} $

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