(1/2x+20)+(3x-85)=x

Solution for (1/2x+20)+(3x-85)=x equation:

(1/2x+20)+(3x-85)=x

We move all terms to the left:

(1/2x+20)+(3x-85)-(x)=0


Domain of the equation: 2x+20)!=0

x∈R


We add all the numbers together, and all the variables

-1x+(1/2x+20)+(3x-85)=0

We get rid of parentheses

-1x+1/2x+3x+20-85=0

We multiply all the terms by the denominator


-1x*2x+3x*2x+20*2x-85*2x+1=0

Wy multiply elements

-2x^2+6x^2+40x-170x+1=0

We add all the numbers together, and all the variables

4x^2-130x+1=0

a = 4; b = -130; c = +1;
Δ = b2-4ac
Δ = -1302-4·4·1
Δ = 16884
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:

$\sqrt{\Delta}=\sqrt{16884}=\sqrt{36*469}=\sqrt{36}*\sqrt{469}=6\sqrt{469}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-130)-6\sqrt{469}}{2*4}=\frac{130-6\sqrt{469}}{8}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-130)+6\sqrt{469}}{2*4}=\frac{130+6\sqrt{469}}{8}
$