5n(7n-1)+4=31+3n

Solution for 5n(7n-1)+4=31+3n equation:

5n(7n-1)+4=31+3n

We move all terms to the left:

5n(7n-1)+4-(31+3n)=0

We add all the numbers together, and all the variables

5n(7n-1)-(3n+31)+4=0

We multiply parentheses

35n^2-5n-(3n+31)+4=0

We get rid of parentheses

35n^2-5n-3n-31+4=0

We add all the numbers together, and all the variables

35n^2-8n-27=0

a = 35; b = -8; c = -27;
Δ = b2-4ac
Δ = -82-4·35·(-27)
Δ = 3844
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{3844}=62$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-8)-62}{2*35}=\frac{-54}{70}
=-27/35
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-8)+62}{2*35}=\frac{70}{70}
=1
$