(12x-23)(71-x)=180

Solution for (12x-23)(71-x)=180 equation:

(12x-23)(71-x)=180

We move all terms to the left:

(12x-23)(71-x)-(180)=0

We add all the numbers together, and all the variables

(12x-23)(-1x+71)-180=0

We multiply parentheses ..

(-12x^2+852x+23x-1633)-180=0

We get rid of parentheses

-12x^2+852x+23x-1633-180=0

We add all the numbers together, and all the variables

-12x^2+875x-1813=0

a = -12; b = 875; c = -1813;
Δ = b2-4ac
Δ = 8752-4·(-12)·(-1813)
Δ = 678601
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{678601}=\sqrt{49*13849}=\sqrt{49}*\sqrt{13849}=7\sqrt{13849}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(875)-7\sqrt{13849}}{2*-12}=\frac{-875-7\sqrt{13849}}{-24}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(875)+7\sqrt{13849}}{2*-12}=\frac{-875+7\sqrt{13849}}{-24}
$