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1/2x+3=23x+1
We move all terms to the left:
1/2x+3-(23x+1)=0
Domain of the equation: 2x!=0We get rid of parentheses
x!=0/2
x!=0
x∈R
1/2x-23x-1+3=0
We multiply all the terms by the denominator
-23x*2x-1*2x+3*2x+1=0
Wy multiply elements
-46x^2-2x+6x+1=0
We add all the numbers together, and all the variables
-46x^2+4x+1=0
a = -46; b = 4; c = +1;
Δ = b2-4ac
Δ = 42-4·(-46)·1
Δ = 200
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}$$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$
The end solution:
$\sqrt{\Delta}=\sqrt{200}=\sqrt{100*2}=\sqrt{100}*\sqrt{2}=10\sqrt{2}$$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(4)-10\sqrt{2}}{2*-46}=\frac{-4-10\sqrt{2}}{-92} $$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(4)+10\sqrt{2}}{2*-46}=\frac{-4+10\sqrt{2}}{-92} $
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