-1/8n+6=3n+31

Solution for -1/8n+6=3n+31 equation:

-1/8n+6=3n+31

We move all terms to the left:

-1/8n+6-(3n+31)=0


Domain of the equation: 8n!=0

n!=0/8

n!=0

n∈R


We get rid of parentheses

-1/8n-3n-31+6=0

We multiply all the terms by the denominator


-3n*8n-31*8n+6*8n-1=0

Wy multiply elements

-24n^2-248n+48n-1=0

We add all the numbers together, and all the variables

-24n^2-200n-1=0

a = -24; b = -200; c = -1;
Δ = b2-4ac
Δ = -2002-4·(-24)·(-1)
Δ = 39904
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{39904}=\sqrt{16*2494}=\sqrt{16}*\sqrt{2494}=4\sqrt{2494}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-200)-4\sqrt{2494}}{2*-24}=\frac{200-4\sqrt{2494}}{-48}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-200)+4\sqrt{2494}}{2*-24}=\frac{200+4\sqrt{2494}}{-48}
$