H(2)=x^2+4x-7

Solution for H(2)=x^2+4x-7 equation:

(2)=H^2+4H-7

We move all terms to the left:

(2)-(H^2+4H-7)=0

We get rid of parentheses

-H^2-4H+7+2=0

We add all the numbers together, and all the variables

-1H^2-4H+9=0

a = -1; b = -4; c = +9;
Δ = b2-4ac
Δ = -42-4·(-1)·9
Δ = 52
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:

$\sqrt{\Delta}=\sqrt{52}=\sqrt{4*13}=\sqrt{4}*\sqrt{13}=2\sqrt{13}$
$H_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-4)-2\sqrt{13}}{2*-1}=\frac{4-2\sqrt{13}}{-2}
$
$H_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-4)+2\sqrt{13}}{2*-1}=\frac{4+2\sqrt{13}}{-2}
$