n^2-19n+72=0

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Solution for n^2-19n+72=0 equation:


Simplifying
n2 + -19n + 72 = 0

Reorder the terms:
72 + -19n + n2 = 0

Solving
72 + -19n + n2 = 0

Solving for variable 'n'.

Begin completing the square.

Move the constant term to the right:

Add '-72' to each side of the equation.
72 + -19n + -72 + n2 = 0 + -72

Reorder the terms:
72 + -72 + -19n + n2 = 0 + -72

Combine like terms: 72 + -72 = 0
0 + -19n + n2 = 0 + -72
-19n + n2 = 0 + -72

Combine like terms: 0 + -72 = -72
-19n + n2 = -72

The n term is -19n.  Take half its coefficient (-9.5).
Square it (90.25) and add it to both sides.

Add '90.25' to each side of the equation.
-19n + 90.25 + n2 = -72 + 90.25

Reorder the terms:
90.25 + -19n + n2 = -72 + 90.25

Combine like terms: -72 + 90.25 = 18.25
90.25 + -19n + n2 = 18.25

Factor a perfect square on the left side:
(n + -9.5)(n + -9.5) = 18.25

Calculate the square root of the right side: 4.272001873

Break this problem into two subproblems by setting 
(n + -9.5) equal to 4.272001873 and -4.272001873.

Subproblem 1

n + -9.5 = 4.272001873 Simplifying n + -9.5 = 4.272001873 Reorder the terms: -9.5 + n = 4.272001873 Solving -9.5 + n = 4.272001873 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '9.5' to each side of the equation. -9.5 + 9.5 + n = 4.272001873 + 9.5 Combine like terms: -9.5 + 9.5 = 0.0 0.0 + n = 4.272001873 + 9.5 n = 4.272001873 + 9.5 Combine like terms: 4.272001873 + 9.5 = 13.772001873 n = 13.772001873 Simplifying n = 13.772001873

Subproblem 2

n + -9.5 = -4.272001873 Simplifying n + -9.5 = -4.272001873 Reorder the terms: -9.5 + n = -4.272001873 Solving -9.5 + n = -4.272001873 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '9.5' to each side of the equation. -9.5 + 9.5 + n = -4.272001873 + 9.5 Combine like terms: -9.5 + 9.5 = 0.0 0.0 + n = -4.272001873 + 9.5 n = -4.272001873 + 9.5 Combine like terms: -4.272001873 + 9.5 = 5.227998127 n = 5.227998127 Simplifying n = 5.227998127

Solution

The solution to the problem is based on the solutions from the subproblems. n = {13.772001873, 5.227998127}

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