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2x+5/(x-1)=13
We move all terms to the left:
2x+5/(x-1)-(13)=0
Domain of the equation: (x-1)!=0We multiply all the terms by the denominator
We move all terms containing x to the left, all other terms to the right
x!=1
x∈R
2x*(x-1)-13*(x-1)+5=0
We multiply parentheses
2x^2-2x-13x+13+5=0
We add all the numbers together, and all the variables
2x^2-15x+18=0
a = 2; b = -15; c = +18;
Δ = b2-4ac
Δ = -152-4·2·18
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$$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-15)-9}{2*2}=\frac{6}{4} =1+1/2 $$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-15)+9}{2*2}=\frac{24}{4} =6 $
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