1/3x+3=5/6x-18

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Solution for 1/3x+3=5/6x-18 equation:



1/3x+3=5/6x-18
We move all terms to the left:
1/3x+3-(5/6x-18)=0
Domain of the equation: 3x!=0
x!=0/3
x!=0
x∈R
Domain of the equation: 6x-18)!=0
x∈R
We get rid of parentheses
1/3x-5/6x+18+3=0
We calculate fractions
6x/18x^2+(-15x)/18x^2+18+3=0
We add all the numbers together, and all the variables
6x/18x^2+(-15x)/18x^2+21=0
We multiply all the terms by the denominator
6x+(-15x)+21*18x^2=0
Wy multiply elements
378x^2+6x+(-15x)=0
We get rid of parentheses
378x^2+6x-15x=0
We add all the numbers together, and all the variables
378x^2-9x=0
a = 378; b = -9; c = 0;
Δ = b2-4ac
Δ = -92-4·378·0
Δ = 81
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{81}=9$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-9)-9}{2*378}=\frac{0}{756} =0 $
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-9)+9}{2*378}=\frac{18}{756} =1/42 $

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