3-1/2x=5x+18

Solution for 3-1/2x=5x+18 equation:

3-1/2x=5x+18

We move all terms to the left:

3-1/2x-(5x+18)=0


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We get rid of parentheses

-1/2x-5x-18+3=0

We multiply all the terms by the denominator


-5x*2x-18*2x+3*2x-1=0

Wy multiply elements

-10x^2-36x+6x-1=0

We add all the numbers together, and all the variables

-10x^2-30x-1=0

a = -10; b = -30; c = -1;
Δ = b2-4ac
Δ = -302-4·(-10)·(-1)
Δ = 860
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:

$\sqrt{\Delta}=\sqrt{860}=\sqrt{4*215}=\sqrt{4}*\sqrt{215}=2\sqrt{215}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-30)-2\sqrt{215}}{2*-10}=\frac{30-2\sqrt{215}}{-20}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-30)+2\sqrt{215}}{2*-10}=\frac{30+2\sqrt{215}}{-20}
$