3n^2+4n-147=0

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Solution for 3n^2+4n-147=0 equation:


Simplifying
3n2 + 4n + -147 = 0

Reorder the terms:
-147 + 4n + 3n2 = 0

Solving
-147 + 4n + 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'.
-49 + 1.333333333n + n2 = 0

Move the constant term to the right:

Add '49' to each side of the equation.
-49 + 1.333333333n + 49 + n2 = 0 + 49

Reorder the terms:
-49 + 49 + 1.333333333n + n2 = 0 + 49

Combine like terms: -49 + 49 = 0
0 + 1.333333333n + n2 = 0 + 49
1.333333333n + n2 = 0 + 49

Combine like terms: 0 + 49 = 49
1.333333333n + n2 = 49

The n term is 1.333333333n.  Take half its coefficient (0.6666666665).
Square it (0.4444444442) and add it to both sides.

Add '0.4444444442' to each side of the equation.
1.333333333n + 0.4444444442 + n2 = 49 + 0.4444444442

Reorder the terms:
0.4444444442 + 1.333333333n + n2 = 49 + 0.4444444442

Combine like terms: 49 + 0.4444444442 = 49.4444444442
0.4444444442 + 1.333333333n + n2 = 49.4444444442

Factor a perfect square on the left side:
(n + 0.6666666665)(n + 0.6666666665) = 49.4444444442

Calculate the square root of the right side: 7.03167437

Break this problem into two subproblems by setting 
(n + 0.6666666665) equal to 7.03167437 and -7.03167437.

Subproblem 1

n + 0.6666666665 = 7.03167437 Simplifying n + 0.6666666665 = 7.03167437 Reorder the terms: 0.6666666665 + n = 7.03167437 Solving 0.6666666665 + n = 7.03167437 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.6666666665' to each side of the equation. 0.6666666665 + -0.6666666665 + n = 7.03167437 + -0.6666666665 Combine like terms: 0.6666666665 + -0.6666666665 = 0.0000000000 0.0000000000 + n = 7.03167437 + -0.6666666665 n = 7.03167437 + -0.6666666665 Combine like terms: 7.03167437 + -0.6666666665 = 6.3650077035 n = 6.3650077035 Simplifying n = 6.3650077035

Subproblem 2

n + 0.6666666665 = -7.03167437 Simplifying n + 0.6666666665 = -7.03167437 Reorder the terms: 0.6666666665 + n = -7.03167437 Solving 0.6666666665 + n = -7.03167437 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.6666666665' to each side of the equation. 0.6666666665 + -0.6666666665 + n = -7.03167437 + -0.6666666665 Combine like terms: 0.6666666665 + -0.6666666665 = 0.0000000000 0.0000000000 + n = -7.03167437 + -0.6666666665 n = -7.03167437 + -0.6666666665 Combine like terms: -7.03167437 + -0.6666666665 = -7.6983410365 n = -7.6983410365 Simplifying n = -7.6983410365

Solution

The solution to the problem is based on the solutions from the subproblems. n = {6.3650077035, -7.6983410365}

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