n^2+7n=6

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


Simplifying
n2 + 7n = 6

Reorder the terms:
7n + n2 = 6

Solving
7n + n2 = 6

Solving for variable 'n'.

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

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

Begin completing the square.

Move the constant term to the right:

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

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

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

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

The n term is 7n.  Take half its coefficient (3.5).
Square it (12.25) and add it to both sides.

Add '12.25' to each side of the equation.
7n + 12.25 + n2 = 6 + 12.25

Reorder the terms:
12.25 + 7n + n2 = 6 + 12.25

Combine like terms: 6 + 12.25 = 18.25
12.25 + 7n + n2 = 18.25

Factor a perfect square on the left side:
(n + 3.5)(n + 3.5) = 18.25

Calculate the square root of the right side: 4.272001873

Break this problem into two subproblems by setting 
(n + 3.5) equal to 4.272001873 and -4.272001873.

Subproblem 1

n + 3.5 = 4.272001873 Simplifying n + 3.5 = 4.272001873 Reorder the terms: 3.5 + n = 4.272001873 Solving 3.5 + n = 4.272001873 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + n = 4.272001873 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + n = 4.272001873 + -3.5 n = 4.272001873 + -3.5 Combine like terms: 4.272001873 + -3.5 = 0.772001873 n = 0.772001873 Simplifying n = 0.772001873

Subproblem 2

n + 3.5 = -4.272001873 Simplifying n + 3.5 = -4.272001873 Reorder the terms: 3.5 + n = -4.272001873 Solving 3.5 + n = -4.272001873 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + n = -4.272001873 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + n = -4.272001873 + -3.5 n = -4.272001873 + -3.5 Combine like terms: -4.272001873 + -3.5 = -7.772001873 n = -7.772001873 Simplifying n = -7.772001873

Solution

The solution to the problem is based on the solutions from the subproblems. n = {0.772001873, -7.772001873}

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