3n-16=n^2-6n+3

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Solution for 3n-16=n^2-6n+3 equation:


Simplifying
3n + -16 = n2 + -6n + 3

Reorder the terms:
-16 + 3n = n2 + -6n + 3

Reorder the terms:
-16 + 3n = 3 + -6n + n2

Solving
-16 + 3n = 3 + -6n + n2

Solving for variable 'n'.

Reorder the terms:
-16 + -3 + 3n + 6n + -1n2 = 3 + -6n + n2 + -3 + 6n + -1n2

Combine like terms: -16 + -3 = -19
-19 + 3n + 6n + -1n2 = 3 + -6n + n2 + -3 + 6n + -1n2

Combine like terms: 3n + 6n = 9n
-19 + 9n + -1n2 = 3 + -6n + n2 + -3 + 6n + -1n2

Reorder the terms:
-19 + 9n + -1n2 = 3 + -3 + -6n + 6n + n2 + -1n2

Combine like terms: 3 + -3 = 0
-19 + 9n + -1n2 = 0 + -6n + 6n + n2 + -1n2
-19 + 9n + -1n2 = -6n + 6n + n2 + -1n2

Combine like terms: -6n + 6n = 0
-19 + 9n + -1n2 = 0 + n2 + -1n2
-19 + 9n + -1n2 = n2 + -1n2

Combine like terms: n2 + -1n2 = 0
-19 + 9n + -1n2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
19 + -9n + n2 = 0

Move the constant term to the right:

Add '-19' to each side of the equation.
19 + -9n + -19 + n2 = 0 + -19

Reorder the terms:
19 + -19 + -9n + n2 = 0 + -19

Combine like terms: 19 + -19 = 0
0 + -9n + n2 = 0 + -19
-9n + n2 = 0 + -19

Combine like terms: 0 + -19 = -19
-9n + n2 = -19

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 = -19 + 20.25

Reorder the terms:
20.25 + -9n + n2 = -19 + 20.25

Combine like terms: -19 + 20.25 = 1.25
20.25 + -9n + n2 = 1.25

Factor a perfect square on the left side:
(n + -4.5)(n + -4.5) = 1.25

Calculate the square root of the right side: 1.118033989

Break this problem into two subproblems by setting 
(n + -4.5) equal to 1.118033989 and -1.118033989.

Subproblem 1

n + -4.5 = 1.118033989 Simplifying n + -4.5 = 1.118033989 Reorder the terms: -4.5 + n = 1.118033989 Solving -4.5 + n = 1.118033989 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 = 1.118033989 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + n = 1.118033989 + 4.5 n = 1.118033989 + 4.5 Combine like terms: 1.118033989 + 4.5 = 5.618033989 n = 5.618033989 Simplifying n = 5.618033989

Subproblem 2

n + -4.5 = -1.118033989 Simplifying n + -4.5 = -1.118033989 Reorder the terms: -4.5 + n = -1.118033989 Solving -4.5 + n = -1.118033989 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 = -1.118033989 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + n = -1.118033989 + 4.5 n = -1.118033989 + 4.5 Combine like terms: -1.118033989 + 4.5 = 3.381966011 n = 3.381966011 Simplifying n = 3.381966011

Solution

The solution to the problem is based on the solutions from the subproblems. n = {5.618033989, 3.381966011}

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