n^2+6n+7=0

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Solution for n^2+6n+7=0 equation:


Simplifying
n2 + 6n + 7 = 0

Reorder the terms:
7 + 6n + n2 = 0

Solving
7 + 6n + n2 = 0

Solving for variable 'n'.

Begin completing the square.

Move the constant term to the right:

Add '-7' to each side of the equation.
7 + 6n + -7 + n2 = 0 + -7

Reorder the terms:
7 + -7 + 6n + n2 = 0 + -7

Combine like terms: 7 + -7 = 0
0 + 6n + n2 = 0 + -7
6n + n2 = 0 + -7

Combine like terms: 0 + -7 = -7
6n + n2 = -7

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 = -7 + 9

Reorder the terms:
9 + 6n + n2 = -7 + 9

Combine like terms: -7 + 9 = 2
9 + 6n + n2 = 2

Factor a perfect square on the left side:
(n + 3)(n + 3) = 2

Calculate the square root of the right side: 1.414213562

Break this problem into two subproblems by setting 
(n + 3) equal to 1.414213562 and -1.414213562.

Subproblem 1

n + 3 = 1.414213562 Simplifying n + 3 = 1.414213562 Reorder the terms: 3 + n = 1.414213562 Solving 3 + n = 1.414213562 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 = 1.414213562 + -3 Combine like terms: 3 + -3 = 0 0 + n = 1.414213562 + -3 n = 1.414213562 + -3 Combine like terms: 1.414213562 + -3 = -1.585786438 n = -1.585786438 Simplifying n = -1.585786438

Subproblem 2

n + 3 = -1.414213562 Simplifying n + 3 = -1.414213562 Reorder the terms: 3 + n = -1.414213562 Solving 3 + n = -1.414213562 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 = -1.414213562 + -3 Combine like terms: 3 + -3 = 0 0 + n = -1.414213562 + -3 n = -1.414213562 + -3 Combine like terms: -1.414213562 + -3 = -4.414213562 n = -4.414213562 Simplifying n = -4.414213562

Solution

The solution to the problem is based on the solutions from the subproblems. n = {-1.585786438, -4.414213562}

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