110=k(k+1)

Solution for 110=k(k+1) equation:

110=k(k+1)

We move all terms to the left:

110-(k(k+1))=0


We calculate terms in parentheses: -(k(k+1)), so:

k(k+1)

We multiply parentheses

k^2+k

Back to the equation:

-(k^2+k)


We get rid of parentheses

-k^2-k+110=0

We add all the numbers together, and all the variables

-1k^2-1k+110=0

a = -1; b = -1; c = +110;
Δ = b2-4ac
Δ = -12-4·(-1)·110
Δ = 441
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$k_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$k_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{441}=21$
$k_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-1)-21}{2*-1}=\frac{-20}{-2}
=+10
$
$k_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-1)+21}{2*-1}=\frac{22}{-2}
=-11
$