H=(x+6)(x+5)+(x+3)(x+8)

Solution for H=(x+6)(x+5)+(x+3)(x+8) equation:

=(H+6)(H+5)+(H+3)(H+8)

We move all terms to the left:

-((H+6)(H+5)+(H+3)(H+8))=0

We multiply parentheses ..

-((+H^2+5H+6H+30)+(H+3)(H+8))=0


We calculate terms in parentheses: -((+H^2+5H+6H+30)+(H+3)(H+8)), so:

(+H^2+5H+6H+30)+(H+3)(H+8)

We get rid of parentheses

H^2+5H+6H+(H+3)(H+8)+30

We multiply parentheses ..

H^2+(+H^2+8H+3H+24)+5H+6H+30

We add all the numbers together, and all the variables

H^2+(+H^2+8H+3H+24)+11H+30

We get rid of parentheses

H^2+H^2+8H+3H+11H+24+30

We add all the numbers together, and all the variables

2H^2+22H+54

Back to the equation:

-(2H^2+22H+54)


We get rid of parentheses

-2H^2-22H-54=0

a = -2; b = -22; c = -54;
Δ = b2-4ac
Δ = -222-4·(-2)·(-54)
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{-(-22)-2\sqrt{13}}{2*-2}=\frac{22-2\sqrt{13}}{-4}
$
$H_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-22)+2\sqrt{13}}{2*-2}=\frac{22+2\sqrt{13}}{-4}
$