12n^2=3n+5

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


Simplifying
12n2 = 3n + 5

Reorder the terms:
12n2 = 5 + 3n

Solving
12n2 = 5 + 3n

Solving for variable 'n'.

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

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

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

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

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

Divide each side by '12'.
-0.4166666667 + -0.25n + n2 = 0

Move the constant term to the right:

Add '0.4166666667' to each side of the equation.
-0.4166666667 + -0.25n + 0.4166666667 + n2 = 0 + 0.4166666667

Reorder the terms:
-0.4166666667 + 0.4166666667 + -0.25n + n2 = 0 + 0.4166666667

Combine like terms: -0.4166666667 + 0.4166666667 = 0.0000000000
0.0000000000 + -0.25n + n2 = 0 + 0.4166666667
-0.25n + n2 = 0 + 0.4166666667

Combine like terms: 0 + 0.4166666667 = 0.4166666667
-0.25n + n2 = 0.4166666667

The n term is -0.25n.  Take half its coefficient (-0.125).
Square it (0.015625) and add it to both sides.

Add '0.015625' to each side of the equation.
-0.25n + 0.015625 + n2 = 0.4166666667 + 0.015625

Reorder the terms:
0.015625 + -0.25n + n2 = 0.4166666667 + 0.015625

Combine like terms: 0.4166666667 + 0.015625 = 0.4322916667
0.015625 + -0.25n + n2 = 0.4322916667

Factor a perfect square on the left side:
(n + -0.125)(n + -0.125) = 0.4322916667

Calculate the square root of the right side: 0.65748891

Break this problem into two subproblems by setting 
(n + -0.125) equal to 0.65748891 and -0.65748891.

Subproblem 1

n + -0.125 = 0.65748891 Simplifying n + -0.125 = 0.65748891 Reorder the terms: -0.125 + n = 0.65748891 Solving -0.125 + n = 0.65748891 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.125' to each side of the equation. -0.125 + 0.125 + n = 0.65748891 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + n = 0.65748891 + 0.125 n = 0.65748891 + 0.125 Combine like terms: 0.65748891 + 0.125 = 0.78248891 n = 0.78248891 Simplifying n = 0.78248891

Subproblem 2

n + -0.125 = -0.65748891 Simplifying n + -0.125 = -0.65748891 Reorder the terms: -0.125 + n = -0.65748891 Solving -0.125 + n = -0.65748891 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.125' to each side of the equation. -0.125 + 0.125 + n = -0.65748891 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + n = -0.65748891 + 0.125 n = -0.65748891 + 0.125 Combine like terms: -0.65748891 + 0.125 = -0.53248891 n = -0.53248891 Simplifying n = -0.53248891

Solution

The solution to the problem is based on the solutions from the subproblems. n = {0.78248891, -0.53248891}

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