1/7n-12=5n+5

Solution for 1/7n-12=5n+5 equation:

1/7n-12=5n+5

We move all terms to the left:

1/7n-12-(5n+5)=0


Domain of the equation: 7n!=0

n!=0/7

n!=0

n∈R


We get rid of parentheses

1/7n-5n-5-12=0

We multiply all the terms by the denominator


-5n*7n-5*7n-12*7n+1=0

Wy multiply elements

-35n^2-35n-84n+1=0

We add all the numbers together, and all the variables

-35n^2-119n+1=0

a = -35; b = -119; c = +1;
Δ = b2-4ac
Δ = -1192-4·(-35)·1
Δ = 14301
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:

$\sqrt{\Delta}=\sqrt{14301}=\sqrt{9*1589}=\sqrt{9}*\sqrt{1589}=3\sqrt{1589}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-119)-3\sqrt{1589}}{2*-35}=\frac{119-3\sqrt{1589}}{-70}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-119)+3\sqrt{1589}}{2*-35}=\frac{119+3\sqrt{1589}}{-70}
$