2x+32=1/2x+65

Solution for 2x+32=1/2x+65 equation:

2x+32=1/2x+65

We move all terms to the left:

2x+32-(1/2x+65)=0


Domain of the equation: 2x+65)!=0

x∈R


We get rid of parentheses

2x-1/2x-65+32=0

We multiply all the terms by the denominator


2x*2x-65*2x+32*2x-1=0

Wy multiply elements

4x^2-130x+64x-1=0

We add all the numbers together, and all the variables

4x^2-66x-1=0

a = 4; b = -66; c = -1;
Δ = b2-4ac
Δ = -662-4·4·(-1)
Δ = 4372
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:

$\sqrt{\Delta}=\sqrt{4372}=\sqrt{4*1093}=\sqrt{4}*\sqrt{1093}=2\sqrt{1093}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-66)-2\sqrt{1093}}{2*4}=\frac{66-2\sqrt{1093}}{8}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-66)+2\sqrt{1093}}{2*4}=\frac{66+2\sqrt{1093}}{8}
$