1/2x+(x-35)+x+(x-46)=180

Solution for 1/2x+(x-35)+x+(x-46)=180 equation:

1/2x+(x-35)+x+(x-46)=180

We move all terms to the left:

1/2x+(x-35)+x+(x-46)-(180)=0


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We add all the numbers together, and all the variables

x+1/2x+(x-35)+(x-46)-180=0

We get rid of parentheses

x+1/2x+x+x-35-46-180=0

We multiply all the terms by the denominator


x*2x+x*2x+x*2x-35*2x-46*2x-180*2x+1=0

Wy multiply elements

2x^2+2x^2+2x^2-70x-92x-360x+1=0

We add all the numbers together, and all the variables

6x^2-522x+1=0

a = 6; b = -522; c = +1;
Δ = b2-4ac
Δ = -5222-4·6·1
Δ = 272460
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:

$\sqrt{\Delta}=\sqrt{272460}=\sqrt{4*68115}=\sqrt{4}*\sqrt{68115}=2\sqrt{68115}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-522)-2\sqrt{68115}}{2*6}=\frac{522-2\sqrt{68115}}{12}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-522)+2\sqrt{68115}}{2*6}=\frac{522+2\sqrt{68115}}{12}
$