n^2-9n-31=0

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Solution for n^2-9n-31=0 equation:


Simplifying
n2 + -9n + -31 = 0

Reorder the terms:
-31 + -9n + n2 = 0

Solving
-31 + -9n + n2 = 0

Solving for variable 'n'.

Begin completing the square.

Move the constant term to the right:

Add '31' to each side of the equation.
-31 + -9n + 31 + n2 = 0 + 31

Reorder the terms:
-31 + 31 + -9n + n2 = 0 + 31

Combine like terms: -31 + 31 = 0
0 + -9n + n2 = 0 + 31
-9n + n2 = 0 + 31

Combine like terms: 0 + 31 = 31
-9n + n2 = 31

The n term is -9n.  Take half its coefficient (-4.5).
Square it (20.25) and add it to both sides.

Add '20.25' to each side of the equation.
-9n + 20.25 + n2 = 31 + 20.25

Reorder the terms:
20.25 + -9n + n2 = 31 + 20.25

Combine like terms: 31 + 20.25 = 51.25
20.25 + -9n + n2 = 51.25

Factor a perfect square on the left side:
(n + -4.5)(n + -4.5) = 51.25

Calculate the square root of the right side: 7.158910532

Break this problem into two subproblems by setting 
(n + -4.5) equal to 7.158910532 and -7.158910532.

Subproblem 1

n + -4.5 = 7.158910532 Simplifying n + -4.5 = 7.158910532 Reorder the terms: -4.5 + n = 7.158910532 Solving -4.5 + n = 7.158910532 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + n = 7.158910532 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + n = 7.158910532 + 4.5 n = 7.158910532 + 4.5 Combine like terms: 7.158910532 + 4.5 = 11.658910532 n = 11.658910532 Simplifying n = 11.658910532

Subproblem 2

n + -4.5 = -7.158910532 Simplifying n + -4.5 = -7.158910532 Reorder the terms: -4.5 + n = -7.158910532 Solving -4.5 + n = -7.158910532 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + n = -7.158910532 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + n = -7.158910532 + 4.5 n = -7.158910532 + 4.5 Combine like terms: -7.158910532 + 4.5 = -2.658910532 n = -2.658910532 Simplifying n = -2.658910532

Solution

The solution to the problem is based on the solutions from the subproblems. n = {11.658910532, -2.658910532}

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