H(x)=-x^2+5x+6

Solution for H(x)=-x^2+5x+6 equation:

(H)=-H^2+5H+6

We move all terms to the left:

(H)-(-H^2+5H+6)=0

We get rid of parentheses

H^2-5H+H-6=0

We add all the numbers together, and all the variables

H^2-4H-6=0

a = 1; b = -4; c = -6;
Δ = b2-4ac
Δ = -42-4·1·(-6)
Δ = 40
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$H_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$H_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{40}=\sqrt{4*10}=\sqrt{4}*\sqrt{10}=2\sqrt{10}$
$H_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-4)-2\sqrt{10}}{2*1}=\frac{4-2\sqrt{10}}{2}
$
$H_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-4)+2\sqrt{10}}{2*1}=\frac{4+2\sqrt{10}}{2}
$