3(5n-1)-14n+9=10(n-4)6n-4(n+1)

Solution for 3(5n-1)-14n+9=10(n-4)6n-4(n+1) equation:

3(5n-1)-14n+9=10(n-4)6n-4(n+1)

We move all terms to the left:

3(5n-1)-14n+9-(10(n-4)6n-4(n+1))=0

We add all the numbers together, and all the variables

-14n+3(5n-1)-(10(n-4)6n-4(n+1))+9=0

We multiply parentheses

-14n+15n-(10(n-4)6n-4(n+1))-3+9=0


We calculate terms in parentheses: -(10(n-4)6n-4(n+1)), so:

10(n-4)6n-4(n+1)

We multiply parentheses

60n^2-240n-4n-4

We add all the numbers together, and all the variables

60n^2-244n-4

Back to the equation:

-(60n^2-244n-4)


We add all the numbers together, and all the variables

n-(60n^2-244n-4)+6=0

We get rid of parentheses

-60n^2+n+244n+4+6=0

We add all the numbers together, and all the variables

-60n^2+245n+10=0

a = -60; b = 245; c = +10;
Δ = b2-4ac
Δ = 2452-4·(-60)·10
Δ = 62425
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:

$\sqrt{\Delta}=\sqrt{62425}=\sqrt{25*2497}=\sqrt{25}*\sqrt{2497}=5\sqrt{2497}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(245)-5\sqrt{2497}}{2*-60}=\frac{-245-5\sqrt{2497}}{-120}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(245)+5\sqrt{2497}}{2*-60}=\frac{-245+5\sqrt{2497}}{-120}
$