x+1/2x+30=4x+110

Solution for x+1/2x+30=4x+110 equation:

x+1/2x+30=4x+110

We move all terms to the left:

x+1/2x+30-(4x+110)=0


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We get rid of parentheses

x+1/2x-4x-110+30=0

We multiply all the terms by the denominator


x*2x-4x*2x-110*2x+30*2x+1=0

Wy multiply elements

2x^2-8x^2-220x+60x+1=0

We add all the numbers together, and all the variables

-6x^2-160x+1=0

a = -6; b = -160; c = +1;
Δ = b2-4ac
Δ = -1602-4·(-6)·1
Δ = 25624
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:

$\sqrt{\Delta}=\sqrt{25624}=\sqrt{4*6406}=\sqrt{4}*\sqrt{6406}=2\sqrt{6406}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-160)-2\sqrt{6406}}{2*-6}=\frac{160-2\sqrt{6406}}{-12}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-160)+2\sqrt{6406}}{2*-6}=\frac{160+2\sqrt{6406}}{-12}
$