3n^2-103n+94=0

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Solution for 3n^2-103n+94=0 equation:


Simplifying
3n2 + -103n + 94 = 0

Reorder the terms:
94 + -103n + 3n2 = 0

Solving
94 + -103n + 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'.
31.33333333 + -34.33333333n + n2 = 0

Move the constant term to the right:

Add '-31.33333333' to each side of the equation.
31.33333333 + -34.33333333n + -31.33333333 + n2 = 0 + -31.33333333

Reorder the terms:
31.33333333 + -31.33333333 + -34.33333333n + n2 = 0 + -31.33333333

Combine like terms: 31.33333333 + -31.33333333 = 0.00000000
0.00000000 + -34.33333333n + n2 = 0 + -31.33333333
-34.33333333n + n2 = 0 + -31.33333333

Combine like terms: 0 + -31.33333333 = -31.33333333
-34.33333333n + n2 = -31.33333333

The n term is -34.33333333n.  Take half its coefficient (-17.16666667).
Square it (294.6944446) and add it to both sides.

Add '294.6944446' to each side of the equation.
-34.33333333n + 294.6944446 + n2 = -31.33333333 + 294.6944446

Reorder the terms:
294.6944446 + -34.33333333n + n2 = -31.33333333 + 294.6944446

Combine like terms: -31.33333333 + 294.6944446 = 263.36111127
294.6944446 + -34.33333333n + n2 = 263.36111127

Factor a perfect square on the left side:
(n + -17.16666667)(n + -17.16666667) = 263.36111127

Calculate the square root of the right side: 16.228404459

Break this problem into two subproblems by setting 
(n + -17.16666667) equal to 16.228404459 and -16.228404459.

Subproblem 1

n + -17.16666667 = 16.228404459 Simplifying n + -17.16666667 = 16.228404459 Reorder the terms: -17.16666667 + n = 16.228404459 Solving -17.16666667 + n = 16.228404459 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '17.16666667' to each side of the equation. -17.16666667 + 17.16666667 + n = 16.228404459 + 17.16666667 Combine like terms: -17.16666667 + 17.16666667 = 0.00000000 0.00000000 + n = 16.228404459 + 17.16666667 n = 16.228404459 + 17.16666667 Combine like terms: 16.228404459 + 17.16666667 = 33.395071129 n = 33.395071129 Simplifying n = 33.395071129

Subproblem 2

n + -17.16666667 = -16.228404459 Simplifying n + -17.16666667 = -16.228404459 Reorder the terms: -17.16666667 + n = -16.228404459 Solving -17.16666667 + n = -16.228404459 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '17.16666667' to each side of the equation. -17.16666667 + 17.16666667 + n = -16.228404459 + 17.16666667 Combine like terms: -17.16666667 + 17.16666667 = 0.00000000 0.00000000 + n = -16.228404459 + 17.16666667 n = -16.228404459 + 17.16666667 Combine like terms: -16.228404459 + 17.16666667 = 0.938262211 n = 0.938262211 Simplifying n = 0.938262211

Solution

The solution to the problem is based on the solutions from the subproblems. n = {33.395071129, 0.938262211}

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