(n-7)(n-7)=6

Solution for (n-7)(n-7)=6 equation:

(n-7)(n-7)=6

We move all terms to the left:

(n-7)(n-7)-(6)=0

We multiply parentheses ..

(+n^2-7n-7n+49)-6=0

We get rid of parentheses

n^2-7n-7n+49-6=0

We add all the numbers together, and all the variables

n^2-14n+43=0

a = 1; b = -14; c = +43;
Δ = b2-4ac
Δ = -142-4·1·43
Δ = 24
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:

$\sqrt{\Delta}=\sqrt{24}=\sqrt{4*6}=\sqrt{4}*\sqrt{6}=2\sqrt{6}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-14)-2\sqrt{6}}{2*1}=\frac{14-2\sqrt{6}}{2}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-14)+2\sqrt{6}}{2*1}=\frac{14+2\sqrt{6}}{2}
$