1/2n-4=14-10n

Solution for 1/2n-4=14-10n equation:

1/2n-4=14-10n

We move all terms to the left:

1/2n-4-(14-10n)=0


Domain of the equation: 2n!=0

n!=0/2

n!=0

n∈R


We add all the numbers together, and all the variables

1/2n-(-10n+14)-4=0

We get rid of parentheses

1/2n+10n-14-4=0

We multiply all the terms by the denominator


10n*2n-14*2n-4*2n+1=0

Wy multiply elements

20n^2-28n-8n+1=0

We add all the numbers together, and all the variables

20n^2-36n+1=0

a = 20; b = -36; c = +1;
Δ = b2-4ac
Δ = -362-4·20·1
Δ = 1216
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:

$\sqrt{\Delta}=\sqrt{1216}=\sqrt{64*19}=\sqrt{64}*\sqrt{19}=8\sqrt{19}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-36)-8\sqrt{19}}{2*20}=\frac{36-8\sqrt{19}}{40}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-36)+8\sqrt{19}}{2*20}=\frac{36+8\sqrt{19}}{40}
$