6n^2-3n=14

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Solution for 6n^2-3n=14 equation:


Simplifying
6n2 + -3n = 14

Reorder the terms:
-3n + 6n2 = 14

Solving
-3n + 6n2 = 14

Solving for variable 'n'.

Reorder the terms:
-14 + -3n + 6n2 = 14 + -14

Combine like terms: 14 + -14 = 0
-14 + -3n + 6n2 = 0

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

Divide each side by '6'.
-2.333333333 + -0.5n + n2 = 0

Move the constant term to the right:

Add '2.333333333' to each side of the equation.
-2.333333333 + -0.5n + 2.333333333 + n2 = 0 + 2.333333333

Reorder the terms:
-2.333333333 + 2.333333333 + -0.5n + n2 = 0 + 2.333333333

Combine like terms: -2.333333333 + 2.333333333 = 0.000000000
0.000000000 + -0.5n + n2 = 0 + 2.333333333
-0.5n + n2 = 0 + 2.333333333

Combine like terms: 0 + 2.333333333 = 2.333333333
-0.5n + n2 = 2.333333333

The n term is -0.5n.  Take half its coefficient (-0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
-0.5n + 0.0625 + n2 = 2.333333333 + 0.0625

Reorder the terms:
0.0625 + -0.5n + n2 = 2.333333333 + 0.0625

Combine like terms: 2.333333333 + 0.0625 = 2.395833333
0.0625 + -0.5n + n2 = 2.395833333

Factor a perfect square on the left side:
(n + -0.25)(n + -0.25) = 2.395833333

Calculate the square root of the right side: 1.547847968

Break this problem into two subproblems by setting 
(n + -0.25) equal to 1.547847968 and -1.547847968.

Subproblem 1

n + -0.25 = 1.547847968 Simplifying n + -0.25 = 1.547847968 Reorder the terms: -0.25 + n = 1.547847968 Solving -0.25 + n = 1.547847968 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + n = 1.547847968 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + n = 1.547847968 + 0.25 n = 1.547847968 + 0.25 Combine like terms: 1.547847968 + 0.25 = 1.797847968 n = 1.797847968 Simplifying n = 1.797847968

Subproblem 2

n + -0.25 = -1.547847968 Simplifying n + -0.25 = -1.547847968 Reorder the terms: -0.25 + n = -1.547847968 Solving -0.25 + n = -1.547847968 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + n = -1.547847968 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + n = -1.547847968 + 0.25 n = -1.547847968 + 0.25 Combine like terms: -1.547847968 + 0.25 = -1.297847968 n = -1.297847968 Simplifying n = -1.297847968

Solution

The solution to the problem is based on the solutions from the subproblems. n = {1.797847968, -1.297847968}

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