1/2x+x=30+180

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

1/2x+x=30+180

We move all terms to the left:

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


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We add all the numbers together, and all the variables

1/2x+x-210=0

We add all the numbers together, and all the variables

x+1/2x-210=0

We multiply all the terms by the denominator


x*2x-210*2x+1=0

Wy multiply elements

2x^2-420x+1=0

a = 2; b = -420; c = +1;
Δ = b2-4ac
Δ = -4202-4·2·1
Δ = 176392
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:

$\sqrt{\Delta}=\sqrt{176392}=\sqrt{4*44098}=\sqrt{4}*\sqrt{44098}=2\sqrt{44098}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-420)-2\sqrt{44098}}{2*2}=\frac{420-2\sqrt{44098}}{4}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-420)+2\sqrt{44098}}{2*2}=\frac{420+2\sqrt{44098}}{4}
$