(x-9)+1/2x=180

Solution for (x-9)+1/2x=180 equation:

(x-9)+1/2x=180

We move all terms to the left:

(x-9)+1/2x-(180)=0


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We get rid of parentheses

x+1/2x-9-180=0

We multiply all the terms by the denominator


x*2x-9*2x-180*2x+1=0

Wy multiply elements

2x^2-18x-360x+1=0

We add all the numbers together, and all the variables

2x^2-378x+1=0

a = 2; b = -378; c = +1;
Δ = b2-4ac
Δ = -3782-4·2·1
Δ = 142876
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:

$\sqrt{\Delta}=\sqrt{142876}=\sqrt{4*35719}=\sqrt{4}*\sqrt{35719}=2\sqrt{35719}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-378)-2\sqrt{35719}}{2*2}=\frac{378-2\sqrt{35719}}{4}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-378)+2\sqrt{35719}}{2*2}=\frac{378+2\sqrt{35719}}{4}
$