x-31=1/2x+32

Solution for x-31=1/2x+32 equation:

x-31=1/2x+32

We move all terms to the left:

x-31-(1/2x+32)=0


Domain of the equation: 2x+32)!=0

x∈R


We get rid of parentheses

x-1/2x-32-31=0

We multiply all the terms by the denominator


x*2x-32*2x-31*2x-1=0

Wy multiply elements

2x^2-64x-62x-1=0

We add all the numbers together, and all the variables

2x^2-126x-1=0

a = 2; b = -126; c = -1;
Δ = b2-4ac
Δ = -1262-4·2·(-1)
Δ = 15884
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:

$\sqrt{\Delta}=\sqrt{15884}=\sqrt{1444*11}=\sqrt{1444}*\sqrt{11}=38\sqrt{11}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-126)-38\sqrt{11}}{2*2}=\frac{126-38\sqrt{11}}{4}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-126)+38\sqrt{11}}{2*2}=\frac{126+38\sqrt{11}}{4}
$