5n^2+107n+72=0

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Solution for 5n^2+107n+72=0 equation:


Simplifying
5n2 + 107n + 72 = 0

Reorder the terms:
72 + 107n + 5n2 = 0

Solving
72 + 107n + 5n2 = 0

Solving for variable 'n'.

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

Divide each side by '5'.
14.4 + 21.4n + n2 = 0

Move the constant term to the right:

Add '-14.4' to each side of the equation.
14.4 + 21.4n + -14.4 + n2 = 0 + -14.4

Reorder the terms:
14.4 + -14.4 + 21.4n + n2 = 0 + -14.4

Combine like terms: 14.4 + -14.4 = 0.0
0.0 + 21.4n + n2 = 0 + -14.4
21.4n + n2 = 0 + -14.4

Combine like terms: 0 + -14.4 = -14.4
21.4n + n2 = -14.4

The n term is 21.4n.  Take half its coefficient (10.7).
Square it (114.49) and add it to both sides.

Add '114.49' to each side of the equation.
21.4n + 114.49 + n2 = -14.4 + 114.49

Reorder the terms:
114.49 + 21.4n + n2 = -14.4 + 114.49

Combine like terms: -14.4 + 114.49 = 100.09
114.49 + 21.4n + n2 = 100.09

Factor a perfect square on the left side:
(n + 10.7)(n + 10.7) = 100.09

Calculate the square root of the right side: 10.004498988

Break this problem into two subproblems by setting 
(n + 10.7) equal to 10.004498988 and -10.004498988.

Subproblem 1

n + 10.7 = 10.004498988 Simplifying n + 10.7 = 10.004498988 Reorder the terms: 10.7 + n = 10.004498988 Solving 10.7 + n = 10.004498988 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-10.7' to each side of the equation. 10.7 + -10.7 + n = 10.004498988 + -10.7 Combine like terms: 10.7 + -10.7 = 0.0 0.0 + n = 10.004498988 + -10.7 n = 10.004498988 + -10.7 Combine like terms: 10.004498988 + -10.7 = -0.695501012 n = -0.695501012 Simplifying n = -0.695501012

Subproblem 2

n + 10.7 = -10.004498988 Simplifying n + 10.7 = -10.004498988 Reorder the terms: 10.7 + n = -10.004498988 Solving 10.7 + n = -10.004498988 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-10.7' to each side of the equation. 10.7 + -10.7 + n = -10.004498988 + -10.7 Combine like terms: 10.7 + -10.7 = 0.0 0.0 + n = -10.004498988 + -10.7 n = -10.004498988 + -10.7 Combine like terms: -10.004498988 + -10.7 = -20.704498988 n = -20.704498988 Simplifying n = -20.704498988

Solution

The solution to the problem is based on the solutions from the subproblems. n = {-0.695501012, -20.704498988}

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