1/9a=3a-5

Solution for 1/9a=3a-5 equation:

1/9a=3a-5

We move all terms to the left:

1/9a-(3a-5)=0


Domain of the equation: 9a!=0

a!=0/9

a!=0

a∈R


We get rid of parentheses

1/9a-3a+5=0

We multiply all the terms by the denominator


-3a*9a+5*9a+1=0

Wy multiply elements

-27a^2+45a+1=0

a = -27; b = 45; c = +1;
Δ = b2-4ac
Δ = 452-4·(-27)·1
Δ = 2133
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:

$\sqrt{\Delta}=\sqrt{2133}=\sqrt{9*237}=\sqrt{9}*\sqrt{237}=3\sqrt{237}$
$a_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(45)-3\sqrt{237}}{2*-27}=\frac{-45-3\sqrt{237}}{-54}
$
$a_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(45)+3\sqrt{237}}{2*-27}=\frac{-45+3\sqrt{237}}{-54}
$