3n^2-3n-4=0

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Solution for 3n^2-3n-4=0 equation:


Simplifying
3n2 + -3n + -4 = 0

Reorder the terms:
-4 + -3n + 3n2 = 0

Solving
-4 + -3n + 3n2 = 0

Solving for variable 'n'.

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

Divide each side by '3'.
-1.333333333 + -1n + n2 = 0

Move the constant term to the right:

Add '1.333333333' to each side of the equation.
-1.333333333 + -1n + 1.333333333 + n2 = 0 + 1.333333333

Reorder the terms:
-1.333333333 + 1.333333333 + -1n + n2 = 0 + 1.333333333

Combine like terms: -1.333333333 + 1.333333333 = 0.000000000
0.000000000 + -1n + n2 = 0 + 1.333333333
-1n + n2 = 0 + 1.333333333

Combine like terms: 0 + 1.333333333 = 1.333333333
-1n + n2 = 1.333333333

The n term is -1n.  Take half its coefficient (-0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
-1n + 0.25 + n2 = 1.333333333 + 0.25

Reorder the terms:
0.25 + -1n + n2 = 1.333333333 + 0.25

Combine like terms: 1.333333333 + 0.25 = 1.583333333
0.25 + -1n + n2 = 1.583333333

Factor a perfect square on the left side:
(n + -0.5)(n + -0.5) = 1.583333333

Calculate the square root of the right side: 1.258305739

Break this problem into two subproblems by setting 
(n + -0.5) equal to 1.258305739 and -1.258305739.

Subproblem 1

n + -0.5 = 1.258305739 Simplifying n + -0.5 = 1.258305739 Reorder the terms: -0.5 + n = 1.258305739 Solving -0.5 + n = 1.258305739 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + n = 1.258305739 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = 1.258305739 + 0.5 n = 1.258305739 + 0.5 Combine like terms: 1.258305739 + 0.5 = 1.758305739 n = 1.758305739 Simplifying n = 1.758305739

Subproblem 2

n + -0.5 = -1.258305739 Simplifying n + -0.5 = -1.258305739 Reorder the terms: -0.5 + n = -1.258305739 Solving -0.5 + n = -1.258305739 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + n = -1.258305739 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = -1.258305739 + 0.5 n = -1.258305739 + 0.5 Combine like terms: -1.258305739 + 0.5 = -0.758305739 n = -0.758305739 Simplifying n = -0.758305739

Solution

The solution to the problem is based on the solutions from the subproblems. n = {1.758305739, -0.758305739}

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