M=n(n+1)=110

Solution for M=n(n+1)=110 equation:

=M(M+1)=110

We move all terms to the left:

-(M(M+1))=0


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

M(M+1)

We multiply parentheses

M^2+M

Back to the equation:

-(M^2+M)


We get rid of parentheses

-M^2-M=0

We add all the numbers together, and all the variables

-1M^2-1M=0

a = -1; b = -1; c = 0;
Δ = b2-4ac
Δ = -12-4·(-1)·0
Δ = 1
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$M_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$M_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{1}=1$
$M_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-1)-1}{2*-1}=\frac{0}{-2}
=0
$
$M_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-1)+1}{2*-1}=\frac{2}{-2}
=-1
$