a(49-a)=180

Solution for a(49-a)=180 equation:

a(49-a)=180

We move all terms to the left:

a(49-a)-(180)=0

We add all the numbers together, and all the variables

a(-1a+49)-180=0

We multiply parentheses

-1a^2+49a-180=0

a = -1; b = 49; c = -180;
Δ = b2-4ac
Δ = 492-4·(-1)·(-180)
Δ = 1681
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$a_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$a_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{1681}=41$
$a_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(49)-41}{2*-1}=\frac{-90}{-2}
=+45
$
$a_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(49)+41}{2*-1}=\frac{-8}{-2}
=+4
$