(3n-47)(n+7)=180

Solution for (3n-47)(n+7)=180 equation:

(3n-47)(n+7)=180

We move all terms to the left:

(3n-47)(n+7)-(180)=0

We multiply parentheses ..

(+3n^2+21n-47n-329)-180=0

We get rid of parentheses

3n^2+21n-47n-329-180=0

We add all the numbers together, and all the variables

3n^2-26n-509=0

a = 3; b = -26; c = -509;
Δ = b2-4ac
Δ = -262-4·3·(-509)
Δ = 6784
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:

$\sqrt{\Delta}=\sqrt{6784}=\sqrt{64*106}=\sqrt{64}*\sqrt{106}=8\sqrt{106}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-26)-8\sqrt{106}}{2*3}=\frac{26-8\sqrt{106}}{6}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-26)+8\sqrt{106}}{2*3}=\frac{26+8\sqrt{106}}{6}
$