n-12=1/2n+91

Solution for n-12=1/2n+91 equation:

n-12=1/2n+91

We move all terms to the left:

n-12-(1/2n+91)=0


Domain of the equation: 2n+91)!=0

n∈R


We get rid of parentheses

n-1/2n-91-12=0

We multiply all the terms by the denominator


n*2n-91*2n-12*2n-1=0

Wy multiply elements

2n^2-182n-24n-1=0

We add all the numbers together, and all the variables

2n^2-206n-1=0

a = 2; b = -206; c = -1;
Δ = b2-4ac
Δ = -2062-4·2·(-1)
Δ = 42444
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:

$\sqrt{\Delta}=\sqrt{42444}=\sqrt{324*131}=\sqrt{324}*\sqrt{131}=18\sqrt{131}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-206)-18\sqrt{131}}{2*2}=\frac{206-18\sqrt{131}}{4}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-206)+18\sqrt{131}}{2*2}=\frac{206+18\sqrt{131}}{4}
$