4n^2+n^2=12n+3

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Solution for 4n^2+n^2=12n+3 equation:


Simplifying
4n2 + n2 = 12n + 3

Combine like terms: 4n2 + n2 = 5n2
5n2 = 12n + 3

Reorder the terms:
5n2 = 3 + 12n

Solving
5n2 = 3 + 12n

Solving for variable 'n'.

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

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

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

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

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

Divide each side by '5'.
-0.6 + -2.4n + n2 = 0

Move the constant term to the right:

Add '0.6' to each side of the equation.
-0.6 + -2.4n + 0.6 + n2 = 0 + 0.6

Reorder the terms:
-0.6 + 0.6 + -2.4n + n2 = 0 + 0.6

Combine like terms: -0.6 + 0.6 = 0.0
0.0 + -2.4n + n2 = 0 + 0.6
-2.4n + n2 = 0 + 0.6

Combine like terms: 0 + 0.6 = 0.6
-2.4n + n2 = 0.6

The n term is -2.4n.  Take half its coefficient (-1.2).
Square it (1.44) and add it to both sides.

Add '1.44' to each side of the equation.
-2.4n + 1.44 + n2 = 0.6 + 1.44

Reorder the terms:
1.44 + -2.4n + n2 = 0.6 + 1.44

Combine like terms: 0.6 + 1.44 = 2.04
1.44 + -2.4n + n2 = 2.04

Factor a perfect square on the left side:
(n + -1.2)(n + -1.2) = 2.04

Calculate the square root of the right side: 1.428285686

Break this problem into two subproblems by setting 
(n + -1.2) equal to 1.428285686 and -1.428285686.

Subproblem 1

n + -1.2 = 1.428285686 Simplifying n + -1.2 = 1.428285686 Reorder the terms: -1.2 + n = 1.428285686 Solving -1.2 + n = 1.428285686 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.2' to each side of the equation. -1.2 + 1.2 + n = 1.428285686 + 1.2 Combine like terms: -1.2 + 1.2 = 0.0 0.0 + n = 1.428285686 + 1.2 n = 1.428285686 + 1.2 Combine like terms: 1.428285686 + 1.2 = 2.628285686 n = 2.628285686 Simplifying n = 2.628285686

Subproblem 2

n + -1.2 = -1.428285686 Simplifying n + -1.2 = -1.428285686 Reorder the terms: -1.2 + n = -1.428285686 Solving -1.2 + n = -1.428285686 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.2' to each side of the equation. -1.2 + 1.2 + n = -1.428285686 + 1.2 Combine like terms: -1.2 + 1.2 = 0.0 0.0 + n = -1.428285686 + 1.2 n = -1.428285686 + 1.2 Combine like terms: -1.428285686 + 1.2 = -0.228285686 n = -0.228285686 Simplifying n = -0.228285686

Solution

The solution to the problem is based on the solutions from the subproblems. n = {2.628285686, -0.228285686}

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