90x-1/2x+90-x=120

Solution for 90x-1/2x+90-x=120 equation:

90x-1/2x+90-x=120

We move all terms to the left:

90x-1/2x+90-x-(120)=0


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We add all the numbers together, and all the variables

89x-1/2x-30=0

We multiply all the terms by the denominator


89x*2x-30*2x-1=0

Wy multiply elements

178x^2-60x-1=0

a = 178; b = -60; c = -1;
Δ = b2-4ac
Δ = -602-4·178·(-1)
Δ = 4312
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:

$\sqrt{\Delta}=\sqrt{4312}=\sqrt{196*22}=\sqrt{196}*\sqrt{22}=14\sqrt{22}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-60)-14\sqrt{22}}{2*178}=\frac{60-14\sqrt{22}}{356}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-60)+14\sqrt{22}}{2*178}=\frac{60+14\sqrt{22}}{356}
$