75/(x-2)=75/x+10

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Solution for 75/(x-2)=75/x+10 equation:



75/(x-2)=75/x+10
We move all terms to the left:
75/(x-2)-(75/x+10)=0
Domain of the equation: (x-2)!=0
We move all terms containing x to the left, all other terms to the right
x!=2
x∈R
Domain of the equation: x+10)!=0
x∈R
We get rid of parentheses
75/(x-2)-75/x-10=0
We calculate fractions
75x/(x^2-2x)+(-75x+150)/(x^2-2x)-10=0
We multiply all the terms by the denominator
75x+(-75x+150)-10*(x^2-2x)=0
We multiply parentheses
-10x^2+75x+(-75x+150)+20x=0
We get rid of parentheses
-10x^2+75x-75x+20x+150=0
We add all the numbers together, and all the variables
-10x^2+20x+150=0
a = -10; b = 20; c = +150;
Δ = b2-4ac
Δ = 202-4·(-10)·150
Δ = 6400
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{6400}=80$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(20)-80}{2*-10}=\frac{-100}{-20} =+5 $
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(20)+80}{2*-10}=\frac{60}{-20} =-3 $

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