n^2-3n-16=0

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Solution for n^2-3n-16=0 equation:


Simplifying
n2 + -3n + -16 = 0

Reorder the terms:
-16 + -3n + n2 = 0

Solving
-16 + -3n + n2 = 0

Solving for variable 'n'.

Begin completing the square.

Move the constant term to the right:

Add '16' to each side of the equation.
-16 + -3n + 16 + n2 = 0 + 16

Reorder the terms:
-16 + 16 + -3n + n2 = 0 + 16

Combine like terms: -16 + 16 = 0
0 + -3n + n2 = 0 + 16
-3n + n2 = 0 + 16

Combine like terms: 0 + 16 = 16
-3n + n2 = 16

The n term is -3n.  Take half its coefficient (-1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
-3n + 2.25 + n2 = 16 + 2.25

Reorder the terms:
2.25 + -3n + n2 = 16 + 2.25

Combine like terms: 16 + 2.25 = 18.25
2.25 + -3n + n2 = 18.25

Factor a perfect square on the left side:
(n + -1.5)(n + -1.5) = 18.25

Calculate the square root of the right side: 4.272001873

Break this problem into two subproblems by setting 
(n + -1.5) equal to 4.272001873 and -4.272001873.

Subproblem 1

n + -1.5 = 4.272001873 Simplifying n + -1.5 = 4.272001873 Reorder the terms: -1.5 + n = 4.272001873 Solving -1.5 + n = 4.272001873 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + n = 4.272001873 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = 4.272001873 + 1.5 n = 4.272001873 + 1.5 Combine like terms: 4.272001873 + 1.5 = 5.772001873 n = 5.772001873 Simplifying n = 5.772001873

Subproblem 2

n + -1.5 = -4.272001873 Simplifying n + -1.5 = -4.272001873 Reorder the terms: -1.5 + n = -4.272001873 Solving -1.5 + n = -4.272001873 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + n = -4.272001873 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = -4.272001873 + 1.5 n = -4.272001873 + 1.5 Combine like terms: -4.272001873 + 1.5 = -2.772001873 n = -2.772001873 Simplifying n = -2.772001873

Solution

The solution to the problem is based on the solutions from the subproblems. n = {5.772001873, -2.772001873}

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