x-38+3x-71=1/2x+19

Solution for x-38+3x-71=1/2x+19 equation:

x-38+3x-71=1/2x+19

We move all terms to the left:

x-38+3x-71-(1/2x+19)=0


Domain of the equation: 2x+19)!=0

x∈R


We add all the numbers together, and all the variables

4x-(1/2x+19)-109=0

We get rid of parentheses

4x-1/2x-19-109=0

We multiply all the terms by the denominator


4x*2x-19*2x-109*2x-1=0

Wy multiply elements

8x^2-38x-218x-1=0

We add all the numbers together, and all the variables

8x^2-256x-1=0

a = 8; b = -256; c = -1;
Δ = b2-4ac
Δ = -2562-4·8·(-1)
Δ = 65568
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:

$\sqrt{\Delta}=\sqrt{65568}=\sqrt{16*4098}=\sqrt{16}*\sqrt{4098}=4\sqrt{4098}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-256)-4\sqrt{4098}}{2*8}=\frac{256-4\sqrt{4098}}{16}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-256)+4\sqrt{4098}}{2*8}=\frac{256+4\sqrt{4098}}{16}
$