1/3y=y+6

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Solution for 1/3y=y+6 equation:



1/3y=y+6
We move all terms to the left:
1/3y-(y+6)=0
Domain of the equation: 3y!=0
y!=0/3
y!=0
y∈R
We get rid of parentheses
1/3y-y-6=0
We multiply all the terms by the denominator
-y*3y-6*3y+1=0
Wy multiply elements
-3y^2-18y+1=0
a = -3; b = -18; c = +1;
Δ = b2-4ac
Δ = -182-4·(-3)·1
Δ = 336
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:
$\sqrt{\Delta}=\sqrt{336}=\sqrt{16*21}=\sqrt{16}*\sqrt{21}=4\sqrt{21}$
$y_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-18)-4\sqrt{21}}{2*-3}=\frac{18-4\sqrt{21}}{-6} $
$y_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-18)+4\sqrt{21}}{2*-3}=\frac{18+4\sqrt{21}}{-6} $

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