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3(x-3)-1/2x=19
We move all terms to the left:
3(x-3)-1/2x-(19)=0
Domain of the equation: 2x!=0We multiply parentheses
x!=0/2
x!=0
x∈R
3x-1/2x-9-19=0
We multiply all the terms by the denominator
3x*2x-9*2x-19*2x-1=0
Wy multiply elements
6x^2-18x-38x-1=0
We add all the numbers together, and all the variables
6x^2-56x-1=0
a = 6; b = -56; c = -1;
Δ = b2-4ac
Δ = -562-4·6·(-1)
Δ = 3160
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}$$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$
The end solution:
$\sqrt{\Delta}=\sqrt{3160}=\sqrt{4*790}=\sqrt{4}*\sqrt{790}=2\sqrt{790}$$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-56)-2\sqrt{790}}{2*6}=\frac{56-2\sqrt{790}}{12} $$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-56)+2\sqrt{790}}{2*6}=\frac{56+2\sqrt{790}}{12} $
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