6n^2-42n+71=0

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Solution for 6n^2-42n+71=0 equation:


Simplifying
6n2 + -42n + 71 = 0

Reorder the terms:
71 + -42n + 6n2 = 0

Solving
71 + -42n + 6n2 = 0

Solving for variable 'n'.

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

Divide each side by '6'.
11.83333333 + -7n + n2 = 0

Move the constant term to the right:

Add '-11.83333333' to each side of the equation.
11.83333333 + -7n + -11.83333333 + n2 = 0 + -11.83333333

Reorder the terms:
11.83333333 + -11.83333333 + -7n + n2 = 0 + -11.83333333

Combine like terms: 11.83333333 + -11.83333333 = 0.00000000
0.00000000 + -7n + n2 = 0 + -11.83333333
-7n + n2 = 0 + -11.83333333

Combine like terms: 0 + -11.83333333 = -11.83333333
-7n + n2 = -11.83333333

The n term is -7n.  Take half its coefficient (-3.5).
Square it (12.25) and add it to both sides.

Add '12.25' to each side of the equation.
-7n + 12.25 + n2 = -11.83333333 + 12.25

Reorder the terms:
12.25 + -7n + n2 = -11.83333333 + 12.25

Combine like terms: -11.83333333 + 12.25 = 0.41666667
12.25 + -7n + n2 = 0.41666667

Factor a perfect square on the left side:
(n + -3.5)(n + -3.5) = 0.41666667

Calculate the square root of the right side: 0.645497227

Break this problem into two subproblems by setting 
(n + -3.5) equal to 0.645497227 and -0.645497227.

Subproblem 1

n + -3.5 = 0.645497227 Simplifying n + -3.5 = 0.645497227 Reorder the terms: -3.5 + n = 0.645497227 Solving -3.5 + n = 0.645497227 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + n = 0.645497227 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + n = 0.645497227 + 3.5 n = 0.645497227 + 3.5 Combine like terms: 0.645497227 + 3.5 = 4.145497227 n = 4.145497227 Simplifying n = 4.145497227

Subproblem 2

n + -3.5 = -0.645497227 Simplifying n + -3.5 = -0.645497227 Reorder the terms: -3.5 + n = -0.645497227 Solving -3.5 + n = -0.645497227 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + n = -0.645497227 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + n = -0.645497227 + 3.5 n = -0.645497227 + 3.5 Combine like terms: -0.645497227 + 3.5 = 2.854502773 n = 2.854502773 Simplifying n = 2.854502773

Solution

The solution to the problem is based on the solutions from the subproblems. n = {4.145497227, 2.854502773}

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