4/7(7n)+10(6-n)=-30

Solution for 4/7(7n)+10(6-n)=-30 equation:

4/7(7n)+10(6-n)=-30

We move all terms to the left:

4/7(7n)+10(6-n)-(-30)=0


Domain of the equation: 77n!=0

n!=0/77

n!=0

n∈R


We add all the numbers together, and all the variables

4/77n+10(-1n+6)-(-30)=0

We add all the numbers together, and all the variables

4/77n+10(-1n+6)+30=0

We multiply parentheses

4/77n-10n+60+30=0

We multiply all the terms by the denominator


-10n*77n+60*77n+30*77n+4=0

Wy multiply elements

-770n^2+4620n+2310n+4=0

We add all the numbers together, and all the variables

-770n^2+6930n+4=0

a = -770; b = 6930; c = +4;
Δ = b2-4ac
Δ = 69302-4·(-770)·4
Δ = 48037220
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:

$\sqrt{\Delta}=\sqrt{48037220}=\sqrt{4*12009305}=\sqrt{4}*\sqrt{12009305}=2\sqrt{12009305}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(6930)-2\sqrt{12009305}}{2*-770}=\frac{-6930-2\sqrt{12009305}}{-1540}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(6930)+2\sqrt{12009305}}{2*-770}=\frac{-6930+2\sqrt{12009305}}{-1540}
$