9-1/2n+5=4n

Solution for 9-1/2n+5=4n equation:

9-1/2n+5=4n

We move all terms to the left:

9-1/2n+5-(4n)=0


Domain of the equation: 2n!=0

n!=0/2

n!=0

n∈R


We add all the numbers together, and all the variables

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

We multiply all the terms by the denominator


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

Wy multiply elements

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

a = -8; b = 28; c = -1;
Δ = b2-4ac
Δ = 282-4·(-8)·(-1)
Δ = 752
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:

$\sqrt{\Delta}=\sqrt{752}=\sqrt{16*47}=\sqrt{16}*\sqrt{47}=4\sqrt{47}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(28)-4\sqrt{47}}{2*-8}=\frac{-28-4\sqrt{47}}{-16}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(28)+4\sqrt{47}}{2*-8}=\frac{-28+4\sqrt{47}}{-16}
$