x+5+1/2x=65

Solution for x+5+1/2x=65 equation:

x+5+1/2x=65

We move all terms to the left:

x+5+1/2x-(65)=0


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We add all the numbers together, and all the variables

x+1/2x-60=0

We multiply all the terms by the denominator


x*2x-60*2x+1=0

Wy multiply elements

2x^2-120x+1=0

a = 2; b = -120; c = +1;
Δ = b2-4ac
Δ = -1202-4·2·1
Δ = 14392
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:

$\sqrt{\Delta}=\sqrt{14392}=\sqrt{4*3598}=\sqrt{4}*\sqrt{3598}=2\sqrt{3598}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-120)-2\sqrt{3598}}{2*2}=\frac{120-2\sqrt{3598}}{4}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-120)+2\sqrt{3598}}{2*2}=\frac{120+2\sqrt{3598}}{4}
$