x+1/2*3*x=5

Solution for x+1/2*3*x=5 equation:

x+1/2*3x=5

We move all terms to the left:

x+1/2*3x-(5)=0


Domain of the equation: 2*3x!=0

x!=0/1

x!=0

x∈R


We multiply all the terms by the denominator


x*2*3x-5*2*3x+1=0

Wy multiply elements

6x^2*3-30x*3+1=0

Wy multiply elements

18x^2-90x+1=0

a = 18; b = -90; c = +1;
Δ = b2-4ac
Δ = -902-4·18·1
Δ = 8028
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:

$\sqrt{\Delta}=\sqrt{8028}=\sqrt{36*223}=\sqrt{36}*\sqrt{223}=6\sqrt{223}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-90)-6\sqrt{223}}{2*18}=\frac{90-6\sqrt{223}}{36}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-90)+6\sqrt{223}}{2*18}=\frac{90+6\sqrt{223}}{36}
$