3/4y+9=54+3y

Solution for 3/4y+9=54+3y equation:

3/4y+9=54+3y

We move all terms to the left:

3/4y+9-(54+3y)=0


Domain of the equation: 4y!=0

y!=0/4

y!=0

y∈R


We add all the numbers together, and all the variables

3/4y-(3y+54)+9=0

We get rid of parentheses

3/4y-3y-54+9=0

We multiply all the terms by the denominator


-3y*4y-54*4y+9*4y+3=0

Wy multiply elements

-12y^2-216y+36y+3=0

We add all the numbers together, and all the variables

-12y^2-180y+3=0

a = -12; b = -180; c = +3;
Δ = b2-4ac
Δ = -1802-4·(-12)·3
Δ = 32544
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$y_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$y_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{32544}=\sqrt{144*226}=\sqrt{144}*\sqrt{226}=12\sqrt{226}$
$y_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-180)-12\sqrt{226}}{2*-12}=\frac{180-12\sqrt{226}}{-24}
$
$y_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-180)+12\sqrt{226}}{2*-12}=\frac{180+12\sqrt{226}}{-24}
$