3n^2-2n-14=4

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Solution for 3n^2-2n-14=4 equation:


Simplifying
3n2 + -2n + -14 = 4

Reorder the terms:
-14 + -2n + 3n2 = 4

Solving
-14 + -2n + 3n2 = 4

Solving for variable 'n'.

Reorder the terms:
-14 + -4 + -2n + 3n2 = 4 + -4

Combine like terms: -14 + -4 = -18
-18 + -2n + 3n2 = 4 + -4

Combine like terms: 4 + -4 = 0
-18 + -2n + 3n2 = 0

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

Divide each side by '3'.
-6 + -0.6666666667n + n2 = 0

Move the constant term to the right:

Add '6' to each side of the equation.
-6 + -0.6666666667n + 6 + n2 = 0 + 6

Reorder the terms:
-6 + 6 + -0.6666666667n + n2 = 0 + 6

Combine like terms: -6 + 6 = 0
0 + -0.6666666667n + n2 = 0 + 6
-0.6666666667n + n2 = 0 + 6

Combine like terms: 0 + 6 = 6
-0.6666666667n + n2 = 6

The n term is -0.6666666667n.  Take half its coefficient (-0.3333333334).
Square it (0.1111111112) and add it to both sides.

Add '0.1111111112' to each side of the equation.
-0.6666666667n + 0.1111111112 + n2 = 6 + 0.1111111112

Reorder the terms:
0.1111111112 + -0.6666666667n + n2 = 6 + 0.1111111112

Combine like terms: 6 + 0.1111111112 = 6.1111111112
0.1111111112 + -0.6666666667n + n2 = 6.1111111112

Factor a perfect square on the left side:
(n + -0.3333333334)(n + -0.3333333334) = 6.1111111112

Calculate the square root of the right side: 2.472066162

Break this problem into two subproblems by setting 
(n + -0.3333333334) equal to 2.472066162 and -2.472066162.

Subproblem 1

n + -0.3333333334 = 2.472066162 Simplifying n + -0.3333333334 = 2.472066162 Reorder the terms: -0.3333333334 + n = 2.472066162 Solving -0.3333333334 + n = 2.472066162 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.3333333334' to each side of the equation. -0.3333333334 + 0.3333333334 + n = 2.472066162 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + n = 2.472066162 + 0.3333333334 n = 2.472066162 + 0.3333333334 Combine like terms: 2.472066162 + 0.3333333334 = 2.8053994954 n = 2.8053994954 Simplifying n = 2.8053994954

Subproblem 2

n + -0.3333333334 = -2.472066162 Simplifying n + -0.3333333334 = -2.472066162 Reorder the terms: -0.3333333334 + n = -2.472066162 Solving -0.3333333334 + n = -2.472066162 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.3333333334' to each side of the equation. -0.3333333334 + 0.3333333334 + n = -2.472066162 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + n = -2.472066162 + 0.3333333334 n = -2.472066162 + 0.3333333334 Combine like terms: -2.472066162 + 0.3333333334 = -2.1387328286 n = -2.1387328286 Simplifying n = -2.1387328286

Solution

The solution to the problem is based on the solutions from the subproblems. n = {2.8053994954, -2.1387328286}

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