2-1/2n=3n+17

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Solution for 2-1/2n=3n+17 equation:



2-1/2n=3n+17
We move all terms to the left:
2-1/2n-(3n+17)=0
Domain of the equation: 2n!=0
n!=0/2
n!=0
n∈R
We get rid of parentheses
-1/2n-3n-17+2=0
We multiply all the terms by the denominator
-3n*2n-17*2n+2*2n-1=0
Wy multiply elements
-6n^2-34n+4n-1=0
We add all the numbers together, and all the variables
-6n^2-30n-1=0
a = -6; b = -30; c = -1;
Δ = b2-4ac
Δ = -302-4·(-6)·(-1)
Δ = 876
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:
$\sqrt{\Delta}=\sqrt{876}=\sqrt{4*219}=\sqrt{4}*\sqrt{219}=2\sqrt{219}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-30)-2\sqrt{219}}{2*-6}=\frac{30-2\sqrt{219}}{-12} $
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-30)+2\sqrt{219}}{2*-6}=\frac{30+2\sqrt{219}}{-12} $

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