690=a+1/9*a

Solution for 690=a+1/9*a equation:

690=a+1/9a

We move all terms to the left:

690-(a+1/9a)=0


Domain of the equation: 9a)!=0

a!=0/1

a!=0

a∈R


We add all the numbers together, and all the variables

-(+a+1/9a)+690=0

We get rid of parentheses

-a-1/9a+690=0

We multiply all the terms by the denominator


-a*9a+690*9a-1=0

Wy multiply elements

-9a^2+6210a-1=0

a = -9; b = 6210; c = -1;
Δ = b2-4ac
Δ = 62102-4·(-9)·(-1)
Δ = 38564064
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:

$\sqrt{\Delta}=\sqrt{38564064}=\sqrt{144*267806}=\sqrt{144}*\sqrt{267806}=12\sqrt{267806}$
$a_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(6210)-12\sqrt{267806}}{2*-9}=\frac{-6210-12\sqrt{267806}}{-18}
$
$a_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(6210)+12\sqrt{267806}}{2*-9}=\frac{-6210+12\sqrt{267806}}{-18}
$