3n^2-54=-3n^2+1+12n

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Solution for 3n^2-54=-3n^2+1+12n equation:


Simplifying
3n2 + -54 = -3n2 + 1 + 12n

Reorder the terms:
-54 + 3n2 = -3n2 + 1 + 12n

Reorder the terms:
-54 + 3n2 = 1 + 12n + -3n2

Solving
-54 + 3n2 = 1 + 12n + -3n2

Solving for variable 'n'.

Reorder the terms:
-54 + -1 + -12n + 3n2 + 3n2 = 1 + 12n + -3n2 + -1 + -12n + 3n2

Combine like terms: -54 + -1 = -55
-55 + -12n + 3n2 + 3n2 = 1 + 12n + -3n2 + -1 + -12n + 3n2

Combine like terms: 3n2 + 3n2 = 6n2
-55 + -12n + 6n2 = 1 + 12n + -3n2 + -1 + -12n + 3n2

Reorder the terms:
-55 + -12n + 6n2 = 1 + -1 + 12n + -12n + -3n2 + 3n2

Combine like terms: 1 + -1 = 0
-55 + -12n + 6n2 = 0 + 12n + -12n + -3n2 + 3n2
-55 + -12n + 6n2 = 12n + -12n + -3n2 + 3n2

Combine like terms: 12n + -12n = 0
-55 + -12n + 6n2 = 0 + -3n2 + 3n2
-55 + -12n + 6n2 = -3n2 + 3n2

Combine like terms: -3n2 + 3n2 = 0
-55 + -12n + 6n2 = 0

Begin completing the square.  Divide all terms by
6 the coefficient of the squared term: 

Divide each side by '6'.
-9.166666667 + -2n + n2 = 0

Move the constant term to the right:

Add '9.166666667' to each side of the equation.
-9.166666667 + -2n + 9.166666667 + n2 = 0 + 9.166666667

Reorder the terms:
-9.166666667 + 9.166666667 + -2n + n2 = 0 + 9.166666667

Combine like terms: -9.166666667 + 9.166666667 = 0.000000000
0.000000000 + -2n + n2 = 0 + 9.166666667
-2n + n2 = 0 + 9.166666667

Combine like terms: 0 + 9.166666667 = 9.166666667
-2n + n2 = 9.166666667

The n term is -2n.  Take half its coefficient (-1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
-2n + 1 + n2 = 9.166666667 + 1

Reorder the terms:
1 + -2n + n2 = 9.166666667 + 1

Combine like terms: 9.166666667 + 1 = 10.166666667
1 + -2n + n2 = 10.166666667

Factor a perfect square on the left side:
(n + -1)(n + -1) = 10.166666667

Calculate the square root of the right side: 3.188521078

Break this problem into two subproblems by setting 
(n + -1) equal to 3.188521078 and -3.188521078.

Subproblem 1

n + -1 = 3.188521078 Simplifying n + -1 = 3.188521078 Reorder the terms: -1 + n = 3.188521078 Solving -1 + n = 3.188521078 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + n = 3.188521078 + 1 Combine like terms: -1 + 1 = 0 0 + n = 3.188521078 + 1 n = 3.188521078 + 1 Combine like terms: 3.188521078 + 1 = 4.188521078 n = 4.188521078 Simplifying n = 4.188521078

Subproblem 2

n + -1 = -3.188521078 Simplifying n + -1 = -3.188521078 Reorder the terms: -1 + n = -3.188521078 Solving -1 + n = -3.188521078 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + n = -3.188521078 + 1 Combine like terms: -1 + 1 = 0 0 + n = -3.188521078 + 1 n = -3.188521078 + 1 Combine like terms: -3.188521078 + 1 = -2.188521078 n = -2.188521078 Simplifying n = -2.188521078

Solution

The solution to the problem is based on the solutions from the subproblems. n = {4.188521078, -2.188521078}

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