(100+x)(150+x)=18,000

Solution for (100+x)(150+x)=18,000 equation:

(100+x)(150+x)=18.000

We move all terms to the left:

(100+x)(150+x)-(18.000)=0

We add all the numbers together, and all the variables

(x+100)(x+150)-18=0

We multiply parentheses ..

(+x^2+150x+100x+15000)-18=0

We get rid of parentheses

x^2+150x+100x+15000-18=0

We add all the numbers together, and all the variables

x^2+250x+14982=0

a = 1; b = 250; c = +14982;
Δ = b2-4ac
Δ = 2502-4·1·14982
Δ = 2572
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:

$\sqrt{\Delta}=\sqrt{2572}=\sqrt{4*643}=\sqrt{4}*\sqrt{643}=2\sqrt{643}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(250)-2\sqrt{643}}{2*1}=\frac{-250-2\sqrt{643}}{2}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(250)+2\sqrt{643}}{2*1}=\frac{-250+2\sqrt{643}}{2}
$