1+x(x+1)=157

Solution for 1+x(x+1)=157 equation:

1+x(x+1)=157

We move all terms to the left:

1+x(x+1)-(157)=0

We add all the numbers together, and all the variables

x(x+1)-156=0

We multiply parentheses

x^2+x-156=0

a = 1; b = 1; c = -156;
Δ = b2-4ac
Δ = 12-4·1·(-156)
Δ = 625
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{625}=25$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(1)-25}{2*1}=\frac{-26}{2}
=-13
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(1)+25}{2*1}=\frac{24}{2}
=12
$