6n^2+29n-5=1

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Solution for 6n^2+29n-5=1 equation:


Simplifying
6n2 + 29n + -5 = 1

Reorder the terms:
-5 + 29n + 6n2 = 1

Solving
-5 + 29n + 6n2 = 1

Solving for variable 'n'.

Reorder the terms:
-5 + -1 + 29n + 6n2 = 1 + -1

Combine like terms: -5 + -1 = -6
-6 + 29n + 6n2 = 1 + -1

Combine like terms: 1 + -1 = 0
-6 + 29n + 6n2 = 0

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

Divide each side by '6'.
-1 + 4.833333333n + n2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + 4.833333333n + 1 + n2 = 0 + 1

Reorder the terms:
-1 + 1 + 4.833333333n + n2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + 4.833333333n + n2 = 0 + 1
4.833333333n + n2 = 0 + 1

Combine like terms: 0 + 1 = 1
4.833333333n + n2 = 1

The n term is 4.833333333n.  Take half its coefficient (2.416666667).
Square it (5.840277779) and add it to both sides.

Add '5.840277779' to each side of the equation.
4.833333333n + 5.840277779 + n2 = 1 + 5.840277779

Reorder the terms:
5.840277779 + 4.833333333n + n2 = 1 + 5.840277779

Combine like terms: 1 + 5.840277779 = 6.840277779
5.840277779 + 4.833333333n + n2 = 6.840277779

Factor a perfect square on the left side:
(n + 2.416666667)(n + 2.416666667) = 6.840277779

Calculate the square root of the right side: 2.615392471

Break this problem into two subproblems by setting 
(n + 2.416666667) equal to 2.615392471 and -2.615392471.

Subproblem 1

n + 2.416666667 = 2.615392471 Simplifying n + 2.416666667 = 2.615392471 Reorder the terms: 2.416666667 + n = 2.615392471 Solving 2.416666667 + n = 2.615392471 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2.416666667' to each side of the equation. 2.416666667 + -2.416666667 + n = 2.615392471 + -2.416666667 Combine like terms: 2.416666667 + -2.416666667 = 0.000000000 0.000000000 + n = 2.615392471 + -2.416666667 n = 2.615392471 + -2.416666667 Combine like terms: 2.615392471 + -2.416666667 = 0.198725804 n = 0.198725804 Simplifying n = 0.198725804

Subproblem 2

n + 2.416666667 = -2.615392471 Simplifying n + 2.416666667 = -2.615392471 Reorder the terms: 2.416666667 + n = -2.615392471 Solving 2.416666667 + n = -2.615392471 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2.416666667' to each side of the equation. 2.416666667 + -2.416666667 + n = -2.615392471 + -2.416666667 Combine like terms: 2.416666667 + -2.416666667 = 0.000000000 0.000000000 + n = -2.615392471 + -2.416666667 n = -2.615392471 + -2.416666667 Combine like terms: -2.615392471 + -2.416666667 = -5.032059138 n = -5.032059138 Simplifying n = -5.032059138

Solution

The solution to the problem is based on the solutions from the subproblems. n = {0.198725804, -5.032059138}

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