1/3a+18=2a-12

Solution for 1/3a+18=2a-12 equation:

1/3a+18=2a-12

We move all terms to the left:

1/3a+18-(2a-12)=0


Domain of the equation: 3a!=0

a!=0/3

a!=0

a∈R


We get rid of parentheses

1/3a-2a+12+18=0

We multiply all the terms by the denominator


-2a*3a+12*3a+18*3a+1=0

Wy multiply elements

-6a^2+36a+54a+1=0

We add all the numbers together, and all the variables

-6a^2+90a+1=0

a = -6; b = 90; c = +1;
Δ = b2-4ac
Δ = 902-4·(-6)·1
Δ = 8124
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$a_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$a_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{8124}=\sqrt{4*2031}=\sqrt{4}*\sqrt{2031}=2\sqrt{2031}$
$a_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(90)-2\sqrt{2031}}{2*-6}=\frac{-90-2\sqrt{2031}}{-12}
$
$a_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(90)+2\sqrt{2031}}{2*-6}=\frac{-90+2\sqrt{2031}}{-12}
$