a+1/6a=42

Solution for a+1/6a=42 equation:

a+1/6a=42

We move all terms to the left:

a+1/6a-(42)=0


Domain of the equation: 6a!=0

a!=0/6

a!=0

a∈R


We multiply all the terms by the denominator


a*6a-42*6a+1=0

Wy multiply elements

6a^2-252a+1=0

a = 6; b = -252; c = +1;
Δ = b2-4ac
Δ = -2522-4·6·1
Δ = 63480
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:

$\sqrt{\Delta}=\sqrt{63480}=\sqrt{2116*30}=\sqrt{2116}*\sqrt{30}=46\sqrt{30}$
$a_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-252)-46\sqrt{30}}{2*6}=\frac{252-46\sqrt{30}}{12}
$
$a_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-252)+46\sqrt{30}}{2*6}=\frac{252+46\sqrt{30}}{12}
$