(n+1)(n+3)=54

Solution for (n+1)(n+3)=54 equation:

(n+1)(n+3)=54

We move all terms to the left:

(n+1)(n+3)-(54)=0

We multiply parentheses ..

(+n^2+3n+n+3)-54=0

We get rid of parentheses

n^2+3n+n+3-54=0

We add all the numbers together, and all the variables

n^2+4n-51=0

a = 1; b = 4; c = -51;
Δ = b2-4ac
Δ = 42-4·1·(-51)
Δ = 220
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:

$\sqrt{\Delta}=\sqrt{220}=\sqrt{4*55}=\sqrt{4}*\sqrt{55}=2\sqrt{55}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(4)-2\sqrt{55}}{2*1}=\frac{-4-2\sqrt{55}}{2}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(4)+2\sqrt{55}}{2*1}=\frac{-4+2\sqrt{55}}{2}
$