n^2+6n-46=0

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Solution for n^2+6n-46=0 equation:


Simplifying
n2 + 6n + -46 = 0

Reorder the terms:
-46 + 6n + n2 = 0

Solving
-46 + 6n + 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 + 6n + 46 + n2 = 0 + 46

Reorder the terms:
-46 + 46 + 6n + n2 = 0 + 46

Combine like terms: -46 + 46 = 0
0 + 6n + n2 = 0 + 46
6n + n2 = 0 + 46

Combine like terms: 0 + 46 = 46
6n + n2 = 46

The n term is 6n.  Take half its coefficient (3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
6n + 9 + n2 = 46 + 9

Reorder the terms:
9 + 6n + n2 = 46 + 9

Combine like terms: 46 + 9 = 55
9 + 6n + n2 = 55

Factor a perfect square on the left side:
(n + 3)(n + 3) = 55

Calculate the square root of the right side: 7.416198487

Break this problem into two subproblems by setting 
(n + 3) equal to 7.416198487 and -7.416198487.

Subproblem 1

n + 3 = 7.416198487 Simplifying n + 3 = 7.416198487 Reorder the terms: 3 + n = 7.416198487 Solving 3 + n = 7.416198487 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + n = 7.416198487 + -3 Combine like terms: 3 + -3 = 0 0 + n = 7.416198487 + -3 n = 7.416198487 + -3 Combine like terms: 7.416198487 + -3 = 4.416198487 n = 4.416198487 Simplifying n = 4.416198487

Subproblem 2

n + 3 = -7.416198487 Simplifying n + 3 = -7.416198487 Reorder the terms: 3 + n = -7.416198487 Solving 3 + n = -7.416198487 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + n = -7.416198487 + -3 Combine like terms: 3 + -3 = 0 0 + n = -7.416198487 + -3 n = -7.416198487 + -3 Combine like terms: -7.416198487 + -3 = -10.416198487 n = -10.416198487 Simplifying n = -10.416198487

Solution

The solution to the problem is based on the solutions from the subproblems. n = {4.416198487, -10.416198487}

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