H(1/2)=-3x^2-11x+4

Solution for H(1/2)=-3x^2-11x+4 equation:

(1/2)=-3H^2-11H+4

We move all terms to the left:

(1/2)-(-3H^2-11H+4)=0

We add all the numbers together, and all the variables

-(-3H^2-11H+4)+(+1/2)=0

We get rid of parentheses

3H^2+11H-4+1/2=0

We multiply all the terms by the denominator


3H^2*2+11H*2+1-4*2=0

We add all the numbers together, and all the variables

3H^2*2+11H*2-7=0

Wy multiply elements

6H^2+22H-7=0

a = 6; b = 22; c = -7;
Δ = b2-4ac
Δ = 222-4·6·(-7)
Δ = 652
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:

$\sqrt{\Delta}=\sqrt{652}=\sqrt{4*163}=\sqrt{4}*\sqrt{163}=2\sqrt{163}$
$H_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(22)-2\sqrt{163}}{2*6}=\frac{-22-2\sqrt{163}}{12}
$
$H_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(22)+2\sqrt{163}}{2*6}=\frac{-22+2\sqrt{163}}{12}
$