7n+3=4/37n+5

Solution for 7n+3=4/37n+5 equation:

7n+3=4/37n+5

We move all terms to the left:

7n+3-(4/37n+5)=0


Domain of the equation: 37n+5)!=0

n∈R


We get rid of parentheses

7n-4/37n-5+3=0

We multiply all the terms by the denominator


7n*37n-5*37n+3*37n-4=0

Wy multiply elements

259n^2-185n+111n-4=0

We add all the numbers together, and all the variables

259n^2-74n-4=0

a = 259; b = -74; c = -4;
Δ = b2-4ac
Δ = -742-4·259·(-4)
Δ = 9620
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:

$\sqrt{\Delta}=\sqrt{9620}=\sqrt{4*2405}=\sqrt{4}*\sqrt{2405}=2\sqrt{2405}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-74)-2\sqrt{2405}}{2*259}=\frac{74-2\sqrt{2405}}{518}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-74)+2\sqrt{2405}}{2*259}=\frac{74+2\sqrt{2405}}{518}
$