n^2+2n=7

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


Simplifying
n2 + 2n = 7

Reorder the terms:
2n + n2 = 7

Solving
2n + n2 = 7

Solving for variable 'n'.

Reorder the terms:
-7 + 2n + n2 = 7 + -7

Combine like terms: 7 + -7 = 0
-7 + 2n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '7' to each side of the equation.
-7 + 2n + 7 + n2 = 0 + 7

Reorder the terms:
-7 + 7 + 2n + n2 = 0 + 7

Combine like terms: -7 + 7 = 0
0 + 2n + n2 = 0 + 7
2n + n2 = 0 + 7

Combine like terms: 0 + 7 = 7
2n + n2 = 7

The n term is 2n.  Take half its coefficient (1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
2n + 1 + n2 = 7 + 1

Reorder the terms:
1 + 2n + n2 = 7 + 1

Combine like terms: 7 + 1 = 8
1 + 2n + n2 = 8

Factor a perfect square on the left side:
(n + 1)(n + 1) = 8

Calculate the square root of the right side: 2.828427125

Break this problem into two subproblems by setting 
(n + 1) equal to 2.828427125 and -2.828427125.

Subproblem 1

n + 1 = 2.828427125 Simplifying n + 1 = 2.828427125 Reorder the terms: 1 + n = 2.828427125 Solving 1 + n = 2.828427125 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + n = 2.828427125 + -1 Combine like terms: 1 + -1 = 0 0 + n = 2.828427125 + -1 n = 2.828427125 + -1 Combine like terms: 2.828427125 + -1 = 1.828427125 n = 1.828427125 Simplifying n = 1.828427125

Subproblem 2

n + 1 = -2.828427125 Simplifying n + 1 = -2.828427125 Reorder the terms: 1 + n = -2.828427125 Solving 1 + n = -2.828427125 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + n = -2.828427125 + -1 Combine like terms: 1 + -1 = 0 0 + n = -2.828427125 + -1 n = -2.828427125 + -1 Combine like terms: -2.828427125 + -1 = -3.828427125 n = -3.828427125 Simplifying n = -3.828427125

Solution

The solution to the problem is based on the solutions from the subproblems. n = {1.828427125, -3.828427125}

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