x+19=1/2x+29

Solution for x+19=1/2x+29 equation:

x+19=1/2x+29

We move all terms to the left:

x+19-(1/2x+29)=0


Domain of the equation: 2x+29)!=0

x∈R


We get rid of parentheses

x-1/2x-29+19=0

We multiply all the terms by the denominator


x*2x-29*2x+19*2x-1=0

Wy multiply elements

2x^2-58x+38x-1=0

We add all the numbers together, and all the variables

2x^2-20x-1=0

a = 2; b = -20; c = -1;
Δ = b2-4ac
Δ = -202-4·2·(-1)
Δ = 408
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:

$\sqrt{\Delta}=\sqrt{408}=\sqrt{4*102}=\sqrt{4}*\sqrt{102}=2\sqrt{102}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-20)-2\sqrt{102}}{2*2}=\frac{20-2\sqrt{102}}{4}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-20)+2\sqrt{102}}{2*2}=\frac{20+2\sqrt{102}}{4}
$