n^2-5n+6n-51=0

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Solution for n^2-5n+6n-51=0 equation:


Simplifying
n2 + -5n + 6n + -51 = 0

Reorder the terms:
-51 + -5n + 6n + n2 = 0

Combine like terms: -5n + 6n = 1n
-51 + 1n + n2 = 0

Solving
-51 + 1n + n2 = 0

Solving for variable 'n'.

Begin completing the square.

Move the constant term to the right:

Add '51' to each side of the equation.
-51 + 1n + 51 + n2 = 0 + 51

Reorder the terms:
-51 + 51 + 1n + n2 = 0 + 51

Combine like terms: -51 + 51 = 0
0 + 1n + n2 = 0 + 51
1n + n2 = 0 + 51

Combine like terms: 0 + 51 = 51
1n + n2 = 51

The n term is 1n.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
n + 0.25 + n2 = 51 + 0.25

Reorder the terms:
0.25 + n + n2 = 51 + 0.25

Combine like terms: 51 + 0.25 = 51.25
0.25 + n + n2 = 51.25

Factor a perfect square on the left side:
(n + 0.5)(n + 0.5) = 51.25

Calculate the square root of the right side: 7.158910532

Break this problem into two subproblems by setting 
(n + 0.5) equal to 7.158910532 and -7.158910532.

Subproblem 1

n + 0.5 = 7.158910532 Simplifying n + 0.5 = 7.158910532 Reorder the terms: 0.5 + n = 7.158910532 Solving 0.5 + n = 7.158910532 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = 7.158910532 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 7.158910532 + -0.5 n = 7.158910532 + -0.5 Combine like terms: 7.158910532 + -0.5 = 6.658910532 n = 6.658910532 Simplifying n = 6.658910532

Subproblem 2

n + 0.5 = -7.158910532 Simplifying n + 0.5 = -7.158910532 Reorder the terms: 0.5 + n = -7.158910532 Solving 0.5 + n = -7.158910532 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = -7.158910532 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -7.158910532 + -0.5 n = -7.158910532 + -0.5 Combine like terms: -7.158910532 + -0.5 = -7.658910532 n = -7.658910532 Simplifying n = -7.658910532

Solution

The solution to the problem is based on the solutions from the subproblems. n = {6.658910532, -7.658910532}

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