R(p)=-p^2+8p-3

Solution for R(p)=-p^2+8p-3 equation:

(R)=-R^2+8R-3

We move all terms to the left:

(R)-(-R^2+8R-3)=0

We get rid of parentheses

R^2-8R+R+3=0

We add all the numbers together, and all the variables

R^2-7R+3=0

a = 1; b = -7; c = +3;
Δ = b2-4ac
Δ = -72-4·1·3
Δ = 37
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$R_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$R_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$R_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-7)-\sqrt{37}}{2*1}=\frac{7-\sqrt{37}}{2}
$
$R_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-7)+\sqrt{37}}{2*1}=\frac{7+\sqrt{37}}{2}
$