1/2x+3=3x-27

Solution for 1/2x+3=3x-27 equation:

1/2x+3=3x-27

We move all terms to the left:

1/2x+3-(3x-27)=0


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We get rid of parentheses

1/2x-3x+27+3=0

We multiply all the terms by the denominator


-3x*2x+27*2x+3*2x+1=0

Wy multiply elements

-6x^2+54x+6x+1=0

We add all the numbers together, and all the variables

-6x^2+60x+1=0

a = -6; b = 60; c = +1;
Δ = b2-4ac
Δ = 602-4·(-6)·1
Δ = 3624
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:

$\sqrt{\Delta}=\sqrt{3624}=\sqrt{4*906}=\sqrt{4}*\sqrt{906}=2\sqrt{906}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(60)-2\sqrt{906}}{2*-6}=\frac{-60-2\sqrt{906}}{-12}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(60)+2\sqrt{906}}{2*-6}=\frac{-60+2\sqrt{906}}{-12}
$