H(X)=(1/2x-6)(x+3)

Solution for H(X)=(1/2x-6)(x+3) equation:

(H)=(1/2H-6)(H+3)

We move all terms to the left:

(H)-((1/2H-6)(H+3))=0


Domain of the equation: 2H-6)(H+3))!=0

H∈R


We multiply parentheses ..

-((+H^2+3H-6H-18))+H=0


We calculate terms in parentheses: -((+H^2+3H-6H-18)), so:

(+H^2+3H-6H-18)

We get rid of parentheses

H^2+3H-6H-18

We add all the numbers together, and all the variables

H^2-3H-18

Back to the equation:

-(H^2-3H-18)


We add all the numbers together, and all the variables

H-(H^2-3H-18)=0

We get rid of parentheses

-H^2+H+3H+18=0

We add all the numbers together, and all the variables

-1H^2+4H+18=0

a = -1; b = 4; c = +18;
Δ = b2-4ac
Δ = 42-4·(-1)·18
Δ = 88
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:

$\sqrt{\Delta}=\sqrt{88}=\sqrt{4*22}=\sqrt{4}*\sqrt{22}=2\sqrt{22}$
$H_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(4)-2\sqrt{22}}{2*-1}=\frac{-4-2\sqrt{22}}{-2}
$
$H_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(4)+2\sqrt{22}}{2*-1}=\frac{-4+2\sqrt{22}}{-2}
$