3n^2+5=12n

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Solution for 3n^2+5=12n equation:


Simplifying
3n2 + 5 = 12n

Reorder the terms:
5 + 3n2 = 12n

Solving
5 + 3n2 = 12n

Solving for variable 'n'.

Reorder the terms:
5 + -12n + 3n2 = 12n + -12n

Combine like terms: 12n + -12n = 0
5 + -12n + 3n2 = 0

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

Divide each side by '3'.
1.666666667 + -4n + n2 = 0

Move the constant term to the right:

Add '-1.666666667' to each side of the equation.
1.666666667 + -4n + -1.666666667 + n2 = 0 + -1.666666667

Reorder the terms:
1.666666667 + -1.666666667 + -4n + n2 = 0 + -1.666666667

Combine like terms: 1.666666667 + -1.666666667 = 0.000000000
0.000000000 + -4n + n2 = 0 + -1.666666667
-4n + n2 = 0 + -1.666666667

Combine like terms: 0 + -1.666666667 = -1.666666667
-4n + n2 = -1.666666667

The n term is -4n.  Take half its coefficient (-2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
-4n + 4 + n2 = -1.666666667 + 4

Reorder the terms:
4 + -4n + n2 = -1.666666667 + 4

Combine like terms: -1.666666667 + 4 = 2.333333333
4 + -4n + n2 = 2.333333333

Factor a perfect square on the left side:
(n + -2)(n + -2) = 2.333333333

Calculate the square root of the right side: 1.527525232

Break this problem into two subproblems by setting 
(n + -2) equal to 1.527525232 and -1.527525232.

Subproblem 1

n + -2 = 1.527525232 Simplifying n + -2 = 1.527525232 Reorder the terms: -2 + n = 1.527525232 Solving -2 + n = 1.527525232 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + n = 1.527525232 + 2 Combine like terms: -2 + 2 = 0 0 + n = 1.527525232 + 2 n = 1.527525232 + 2 Combine like terms: 1.527525232 + 2 = 3.527525232 n = 3.527525232 Simplifying n = 3.527525232

Subproblem 2

n + -2 = -1.527525232 Simplifying n + -2 = -1.527525232 Reorder the terms: -2 + n = -1.527525232 Solving -2 + n = -1.527525232 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + n = -1.527525232 + 2 Combine like terms: -2 + 2 = 0 0 + n = -1.527525232 + 2 n = -1.527525232 + 2 Combine like terms: -1.527525232 + 2 = 0.472474768 n = 0.472474768 Simplifying n = 0.472474768

Solution

The solution to the problem is based on the solutions from the subproblems. n = {3.527525232, 0.472474768}

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