P(x)=(-0.5x^2+15x)-(80x+160/x+12)

Solution for P(x)=(-0.5x^2+15x)-(80x+160/x+12) equation:

(P)=(-0.5P^2+15P)-(80P+160/P+12)

We move all terms to the left:

(P)-((-0.5P^2+15P)-(80P+160/P+12))=0


Domain of the equation: P+12))!=0

P∈R


We multiply all the terms by the denominator


-((-0.5P^2+15P)-(80P+160+P*P+12))=0


We calculate terms in parentheses: -((-0.5P^2+15P)-(80P+160+P*P+12)), so:

(-0.5P^2+15P)-(80P+160+P*P+12)

We add all the numbers together, and all the variables

(-0.5P^2+15P)-(80P+P*P+172)

We get rid of parentheses

-0.5P^2+15P-80P-P*P-172

We add all the numbers together, and all the variables

-0.5P^2-65P-P*P-172

Wy multiply elements

-0.5P^2-1P^2-65P-172

We add all the numbers together, and all the variables

-1.5P^2-65P-172

Back to the equation:

-(-1.5P^2-65P-172)


We get rid of parentheses

1.5P^2+65P+172=0

a = 1.5; b = 65; c = +172;
Δ = b2-4ac
Δ = 652-4·1.5·172
Δ = 3193
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$P_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$P_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$P_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(65)-\sqrt{3193}}{2*1.5}=\frac{-65-\sqrt{3193}}{3}
$
$P_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(65)+\sqrt{3193}}{2*1.5}=\frac{-65+\sqrt{3193}}{3}
$