x(x+1)=3(x+x+1)+111

Solution for x(x+1)=3(x+x+1)+111 equation:

x(x+1)=3(x+x+1)+111

We move all terms to the left:

x(x+1)-(3(x+x+1)+111)=0

We add all the numbers together, and all the variables

x(x+1)-(3(2x+1)+111)=0

We multiply parentheses

x^2+x-(3(2x+1)+111)=0


We calculate terms in parentheses: -(3(2x+1)+111), so:

3(2x+1)+111

We multiply parentheses

6x+3+111

We add all the numbers together, and all the variables

6x+114

Back to the equation:

-(6x+114)


We get rid of parentheses

x^2+x-6x-114=0

We add all the numbers together, and all the variables

x^2-5x-114=0

a = 1; b = -5; c = -114;
Δ = b2-4ac
Δ = -52-4·1·(-114)
Δ = 481
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-5)-\sqrt{481}}{2*1}=\frac{5-\sqrt{481}}{2}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-5)+\sqrt{481}}{2*1}=\frac{5+\sqrt{481}}{2}
$