H(x)=(-4x-3)(x-3)

Solution for H(x)=(-4x-3)(x-3) equation:

(H)=(-4H-3)(H-3)

We move all terms to the left:

(H)-((-4H-3)(H-3))=0

We multiply parentheses ..

-((-4H^2+12H-3H+9))+H=0


We calculate terms in parentheses: -((-4H^2+12H-3H+9)), so:

(-4H^2+12H-3H+9)

We get rid of parentheses

-4H^2+12H-3H+9

We add all the numbers together, and all the variables

-4H^2+9H+9

Back to the equation:

-(-4H^2+9H+9)


We get rid of parentheses

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

We add all the numbers together, and all the variables

4H^2-8H-9=0

a = 4; b = -8; c = -9;
Δ = b2-4ac
Δ = -82-4·4·(-9)
Δ = 208
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:

$\sqrt{\Delta}=\sqrt{208}=\sqrt{16*13}=\sqrt{16}*\sqrt{13}=4\sqrt{13}$
$H_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-8)-4\sqrt{13}}{2*4}=\frac{8-4\sqrt{13}}{8}
$
$H_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-8)+4\sqrt{13}}{2*4}=\frac{8+4\sqrt{13}}{8}
$