(1/2x)+(x-30)+(x)=180

Solution for (1/2x)+(x-30)+(x)=180 equation:

(1/2x)+(x-30)+(x)=180

We move all terms to the left:

(1/2x)+(x-30)+(x)-(180)=0


Domain of the equation: 2x)!=0

x!=0/1

x!=0

x∈R


We add all the numbers together, and all the variables

(+1/2x)+(x-30)+x-180=0

We add all the numbers together, and all the variables

x+(+1/2x)+(x-30)-180=0

We get rid of parentheses

x+1/2x+x-30-180=0

We multiply all the terms by the denominator


x*2x+x*2x-30*2x-180*2x+1=0

Wy multiply elements

2x^2+2x^2-60x-360x+1=0

We add all the numbers together, and all the variables

4x^2-420x+1=0

a = 4; b = -420; c = +1;
Δ = b2-4ac
Δ = -4202-4·4·1
Δ = 176384
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:

$\sqrt{\Delta}=\sqrt{176384}=\sqrt{256*689}=\sqrt{256}*\sqrt{689}=16\sqrt{689}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-420)-16\sqrt{689}}{2*4}=\frac{420-16\sqrt{689}}{8}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-420)+16\sqrt{689}}{2*4}=\frac{420+16\sqrt{689}}{8}
$