5n+7=7n(n+1)-2n

Solution for 5n+7=7n(n+1)-2n equation:

5n+7=7n(n+1)-2n

We move all terms to the left:

5n+7-(7n(n+1)-2n)=0


We calculate terms in parentheses: -(7n(n+1)-2n), so:

7n(n+1)-2n

We add all the numbers together, and all the variables

-2n+7n(n+1)

We multiply parentheses

7n^2-2n+7n

We add all the numbers together, and all the variables

7n^2+5n

Back to the equation:

-(7n^2+5n)


We get rid of parentheses

-7n^2+5n-5n+7=0

We add all the numbers together, and all the variables

-7n^2+7=0

a = -7; b = 0; c = +7;
Δ = b2-4ac
Δ = 02-4·(-7)·7
Δ = 196
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{196}=14$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(0)-14}{2*-7}=\frac{-14}{-14}
=1
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(0)+14}{2*-7}=\frac{14}{-14}
=-1
$