x+1/2x+(x-5)=180

Solution for x+1/2x+(x-5)=180 equation:

x+1/2x+(x-5)=180

We move all terms to the left:

x+1/2x+(x-5)-(180)=0


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We get rid of parentheses

x+1/2x+x-5-180=0

We multiply all the terms by the denominator


x*2x+x*2x-5*2x-180*2x+1=0

Wy multiply elements

2x^2+2x^2-10x-360x+1=0

We add all the numbers together, and all the variables

4x^2-370x+1=0

a = 4; b = -370; c = +1;
Δ = b2-4ac
Δ = -3702-4·4·1
Δ = 136884
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:

$\sqrt{\Delta}=\sqrt{136884}=\sqrt{4*34221}=\sqrt{4}*\sqrt{34221}=2\sqrt{34221}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-370)-2\sqrt{34221}}{2*4}=\frac{370-2\sqrt{34221}}{8}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-370)+2\sqrt{34221}}{2*4}=\frac{370+2\sqrt{34221}}{8}
$