3(4d-1)-(3d-2)=5/9d

Solution for 3(4d-1)-(3d-2)=5/9d equation:

3(4d-1)-(3d-2)=5/9d

We move all terms to the left:

3(4d-1)-(3d-2)-(5/9d)=0


Domain of the equation: 9d)!=0

d!=0/1

d!=0

d∈R


We add all the numbers together, and all the variables

3(4d-1)-(3d-2)-(+5/9d)=0

We multiply parentheses

12d-(3d-2)-(+5/9d)-3=0

We get rid of parentheses

12d-3d-5/9d+2-3=0

We multiply all the terms by the denominator


12d*9d-3d*9d+2*9d-3*9d-5=0

Wy multiply elements

108d^2-27d^2+18d-27d-5=0

We add all the numbers together, and all the variables

81d^2-9d-5=0

a = 81; b = -9; c = -5;
Δ = b2-4ac
Δ = -92-4·81·(-5)
Δ = 1701
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$d_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$d_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{1701}=\sqrt{81*21}=\sqrt{81}*\sqrt{21}=9\sqrt{21}$
$d_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-9)-9\sqrt{21}}{2*81}=\frac{9-9\sqrt{21}}{162}
$
$d_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-9)+9\sqrt{21}}{2*81}=\frac{9+9\sqrt{21}}{162}
$