x(x+1)=1/3+14

Solution for x(x+1)=1/3+14 equation:

x(x+1)=1/3+14

We move all terms to the left:

x(x+1)-(1/3+14)=0

We multiply parentheses

x^2+x-(1/3+14)=0

We get rid of parentheses

x^2+x-14-1/3=0

We multiply all the terms by the denominator


x^2*3+x*3-1-14*3=0

We add all the numbers together, and all the variables

x^2*3+x*3-43=0

Wy multiply elements

3x^2+3x-43=0

a = 3; b = 3; c = -43;
Δ = b2-4ac
Δ = 32-4·3·(-43)
Δ = 525
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{525}=\sqrt{25*21}=\sqrt{25}*\sqrt{21}=5\sqrt{21}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(3)-5\sqrt{21}}{2*3}=\frac{-3-5\sqrt{21}}{6}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(3)+5\sqrt{21}}{2*3}=\frac{-3+5\sqrt{21}}{6}
$