n=5(2n-3)(9+3n)

Solution for n=5(2n-3)(9+3n) equation:

n=5(2n-3)(9+3n)

We move all terms to the left:

n-(5(2n-3)(9+3n))=0

We add all the numbers together, and all the variables

n-(5(2n-3)(3n+9))=0

We multiply parentheses ..

-(5(+6n^2+18n-9n-27))+n=0


We calculate terms in parentheses: -(5(+6n^2+18n-9n-27)), so:

5(+6n^2+18n-9n-27)

We multiply parentheses

30n^2+90n-45n-135

We add all the numbers together, and all the variables

30n^2+45n-135

Back to the equation:

-(30n^2+45n-135)


We add all the numbers together, and all the variables

n-(30n^2+45n-135)=0

We get rid of parentheses

-30n^2+n-45n+135=0

We add all the numbers together, and all the variables

-30n^2-44n+135=0

a = -30; b = -44; c = +135;
Δ = b2-4ac
Δ = -442-4·(-30)·135
Δ = 18136
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{18136}=\sqrt{4*4534}=\sqrt{4}*\sqrt{4534}=2\sqrt{4534}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-44)-2\sqrt{4534}}{2*-30}=\frac{44-2\sqrt{4534}}{-60}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-44)+2\sqrt{4534}}{2*-30}=\frac{44+2\sqrt{4534}}{-60}
$