H(x)=X^2+4X-45

Solution for H(x)=X^2+4X-45 equation:

(H)=H^2+4H-45

We move all terms to the left:

(H)-(H^2+4H-45)=0

We get rid of parentheses

-H^2+H-4H+45=0

We add all the numbers together, and all the variables

-1H^2-3H+45=0

a = -1; b = -3; c = +45;
Δ = b2-4ac
Δ = -32-4·(-1)·45
Δ = 189
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:

$\sqrt{\Delta}=\sqrt{189}=\sqrt{9*21}=\sqrt{9}*\sqrt{21}=3\sqrt{21}$
$H_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-3)-3\sqrt{21}}{2*-1}=\frac{3-3\sqrt{21}}{-2}
$
$H_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-3)+3\sqrt{21}}{2*-1}=\frac{3+3\sqrt{21}}{-2}
$