500+30(n-3)=350+45(n-5)n=

Solution for 500+30(n-3)=350+45(n-5)n= equation:

500+30(n-3)=350+45(n-5)n=

We move all terms to the left:

500+30(n-3)-(350+45(n-5)n)=0

We multiply parentheses

30n-(350+45(n-5)n)-90+500=0


We calculate terms in parentheses: -(350+45(n-5)n), so:

350+45(n-5)n

determiningTheFunctionDomain
45(n-5)n+350

We multiply parentheses

45n^2-225n+350

Back to the equation:

-(45n^2-225n+350)


We add all the numbers together, and all the variables

30n-(45n^2-225n+350)+410=0

We get rid of parentheses

-45n^2+30n+225n-350+410=0

We add all the numbers together, and all the variables

-45n^2+255n+60=0

a = -45; b = 255; c = +60;
Δ = b2-4ac
Δ = 2552-4·(-45)·60
Δ = 75825
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:

$\sqrt{\Delta}=\sqrt{75825}=\sqrt{225*337}=\sqrt{225}*\sqrt{337}=15\sqrt{337}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(255)-15\sqrt{337}}{2*-45}=\frac{-255-15\sqrt{337}}{-90}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(255)+15\sqrt{337}}{2*-45}=\frac{-255+15\sqrt{337}}{-90}
$