1/2*s=4s-21

Solution for 1/2*s=4s-21 equation:

1/2s=4s-21

We move all terms to the left:

1/2s-(4s-21)=0


Domain of the equation: 2s!=0

s!=0/2

s!=0

s∈R


We get rid of parentheses

1/2s-4s+21=0

We multiply all the terms by the denominator


-4s*2s+21*2s+1=0

Wy multiply elements

-8s^2+42s+1=0

a = -8; b = 42; c = +1;
Δ = b2-4ac
Δ = 422-4·(-8)·1
Δ = 1796
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:

$\sqrt{\Delta}=\sqrt{1796}=\sqrt{4*449}=\sqrt{4}*\sqrt{449}=2\sqrt{449}$
$s_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(42)-2\sqrt{449}}{2*-8}=\frac{-42-2\sqrt{449}}{-16}
$
$s_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(42)+2\sqrt{449}}{2*-8}=\frac{-42+2\sqrt{449}}{-16}
$