n^2-n-46=0

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Solution for n^2-n-46=0 equation:


Simplifying
n2 + -1n + -46 = 0

Reorder the terms:
-46 + -1n + n2 = 0

Solving
-46 + -1n + n2 = 0

Solving for variable 'n'.

Begin completing the square.

Move the constant term to the right:

Add '46' to each side of the equation.
-46 + -1n + 46 + n2 = 0 + 46

Reorder the terms:
-46 + 46 + -1n + n2 = 0 + 46

Combine like terms: -46 + 46 = 0
0 + -1n + n2 = 0 + 46
-1n + n2 = 0 + 46

Combine like terms: 0 + 46 = 46
-1n + n2 = 46

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.
-1n + 0.25 + n2 = 46 + 0.25

Reorder the terms:
0.25 + -1n + n2 = 46 + 0.25

Combine like terms: 46 + 0.25 = 46.25
0.25 + -1n + n2 = 46.25

Factor a perfect square on the left side:
(n + -0.5)(n + -0.5) = 46.25

Calculate the square root of the right side: 6.800735254

Break this problem into two subproblems by setting 
(n + -0.5) equal to 6.800735254 and -6.800735254.

Subproblem 1

n + -0.5 = 6.800735254 Simplifying n + -0.5 = 6.800735254 Reorder the terms: -0.5 + n = 6.800735254 Solving -0.5 + n = 6.800735254 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 = 6.800735254 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = 6.800735254 + 0.5 n = 6.800735254 + 0.5 Combine like terms: 6.800735254 + 0.5 = 7.300735254 n = 7.300735254 Simplifying n = 7.300735254

Subproblem 2

n + -0.5 = -6.800735254 Simplifying n + -0.5 = -6.800735254 Reorder the terms: -0.5 + n = -6.800735254 Solving -0.5 + n = -6.800735254 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 = -6.800735254 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = -6.800735254 + 0.5 n = -6.800735254 + 0.5 Combine like terms: -6.800735254 + 0.5 = -6.300735254 n = -6.300735254 Simplifying n = -6.300735254

Solution

The solution to the problem is based on the solutions from the subproblems. n = {7.300735254, -6.300735254}

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