7n^2+51n+62=0

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Solution for 7n^2+51n+62=0 equation:


Simplifying
7n2 + 51n + 62 = 0

Reorder the terms:
62 + 51n + 7n2 = 0

Solving
62 + 51n + 7n2 = 0

Solving for variable 'n'.

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

Divide each side by '7'.
8.857142857 + 7.285714286n + n2 = 0

Move the constant term to the right:

Add '-8.857142857' to each side of the equation.
8.857142857 + 7.285714286n + -8.857142857 + n2 = 0 + -8.857142857

Reorder the terms:
8.857142857 + -8.857142857 + 7.285714286n + n2 = 0 + -8.857142857

Combine like terms: 8.857142857 + -8.857142857 = 0.000000000
0.000000000 + 7.285714286n + n2 = 0 + -8.857142857
7.285714286n + n2 = 0 + -8.857142857

Combine like terms: 0 + -8.857142857 = -8.857142857
7.285714286n + n2 = -8.857142857

The n term is 7.285714286n.  Take half its coefficient (3.642857143).
Square it (13.27040816) and add it to both sides.

Add '13.27040816' to each side of the equation.
7.285714286n + 13.27040816 + n2 = -8.857142857 + 13.27040816

Reorder the terms:
13.27040816 + 7.285714286n + n2 = -8.857142857 + 13.27040816

Combine like terms: -8.857142857 + 13.27040816 = 4.413265303
13.27040816 + 7.285714286n + n2 = 4.413265303

Factor a perfect square on the left side:
(n + 3.642857143)(n + 3.642857143) = 4.413265303

Calculate the square root of the right side: 2.100777309

Break this problem into two subproblems by setting 
(n + 3.642857143) equal to 2.100777309 and -2.100777309.

Subproblem 1

n + 3.642857143 = 2.100777309 Simplifying n + 3.642857143 = 2.100777309 Reorder the terms: 3.642857143 + n = 2.100777309 Solving 3.642857143 + n = 2.100777309 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-3.642857143' to each side of the equation. 3.642857143 + -3.642857143 + n = 2.100777309 + -3.642857143 Combine like terms: 3.642857143 + -3.642857143 = 0.000000000 0.000000000 + n = 2.100777309 + -3.642857143 n = 2.100777309 + -3.642857143 Combine like terms: 2.100777309 + -3.642857143 = -1.542079834 n = -1.542079834 Simplifying n = -1.542079834

Subproblem 2

n + 3.642857143 = -2.100777309 Simplifying n + 3.642857143 = -2.100777309 Reorder the terms: 3.642857143 + n = -2.100777309 Solving 3.642857143 + n = -2.100777309 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-3.642857143' to each side of the equation. 3.642857143 + -3.642857143 + n = -2.100777309 + -3.642857143 Combine like terms: 3.642857143 + -3.642857143 = 0.000000000 0.000000000 + n = -2.100777309 + -3.642857143 n = -2.100777309 + -3.642857143 Combine like terms: -2.100777309 + -3.642857143 = -5.743634452 n = -5.743634452 Simplifying n = -5.743634452

Solution

The solution to the problem is based on the solutions from the subproblems. n = {-1.542079834, -5.743634452}

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