S+3s+1/2s+20=65

Solution for S+3s+1/2s+20=65 equation:

+3S+1/2S+20=65

We move all terms to the left:

+3S+1/2S+20-(65)=0


Domain of the equation: 2S!=0

S!=0/2

S!=0

S∈R


We add all the numbers together, and all the variables

3S+1/2S-45=0

We multiply all the terms by the denominator


3S*2S-45*2S+1=0

Wy multiply elements

6S^2-90S+1=0

a = 6; b = -90; c = +1;
Δ = b2-4ac
Δ = -902-4·6·1
Δ = 8076
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$S_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$S_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{8076}=\sqrt{4*2019}=\sqrt{4}*\sqrt{2019}=2\sqrt{2019}$
$S_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-90)-2\sqrt{2019}}{2*6}=\frac{90-2\sqrt{2019}}{12}
$
$S_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-90)+2\sqrt{2019}}{2*6}=\frac{90+2\sqrt{2019}}{12}
$