x+2(x+10)+(2x-12)+(1/2x+6)=106

Solution for x+2(x+10)+(2x-12)+(1/2x+6)=106 equation:

x+2(x+10)+(2x-12)+(1/2x+6)=106

We move all terms to the left:

x+2(x+10)+(2x-12)+(1/2x+6)-(106)=0


Domain of the equation: 2x+6)!=0

x∈R


We multiply parentheses

x+2x+(2x-12)+(1/2x+6)+20-106=0

We get rid of parentheses

x+2x+2x+1/2x-12+6+20-106=0

We multiply all the terms by the denominator


x*2x+2x*2x+2x*2x-12*2x+6*2x+20*2x-106*2x+1=0

Wy multiply elements

2x^2+4x^2+4x^2-24x+12x+40x-212x+1=0

We add all the numbers together, and all the variables

10x^2-184x+1=0

a = 10; b = -184; c = +1;
Δ = b2-4ac
Δ = -1842-4·10·1
Δ = 33816
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:

$\sqrt{\Delta}=\sqrt{33816}=\sqrt{4*8454}=\sqrt{4}*\sqrt{8454}=2\sqrt{8454}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-184)-2\sqrt{8454}}{2*10}=\frac{184-2\sqrt{8454}}{20}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-184)+2\sqrt{8454}}{2*10}=\frac{184+2\sqrt{8454}}{20}
$