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5x/6x-18=1/x-3
We move all terms to the left:
5x/6x-18-(1/x-3)=0
Domain of the equation: 6x!=0
x!=0/6
x!=0
x∈R
Domain of the equation: x-3)!=0We get rid of parentheses
x∈R
5x/6x-1/x+3-18=0
We calculate fractions
5x^2/6x^2+(-6x)/6x^2+3-18=0
We add all the numbers together, and all the variables
5x^2/6x^2+(-6x)/6x^2-15=0
We multiply all the terms by the denominator
5x^2+(-6x)-15*6x^2=0
Wy multiply elements
5x^2-90x^2+(-6x)=0
We get rid of parentheses
5x^2-90x^2-6x=0
We add all the numbers together, and all the variables
-85x^2-6x=0
a = -85; b = -6; c = 0;
Δ = b2-4ac
Δ = -62-4·(-85)·0
Δ = 36
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{36}=6$$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-6)-6}{2*-85}=\frac{0}{-170} =0 $$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-6)+6}{2*-85}=\frac{12}{-170} =-6/85 $
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