n^2+2n-51=0

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Solution for n^2+2n-51=0 equation:


Simplifying
n2 + 2n + -51 = 0

Reorder the terms:
-51 + 2n + n2 = 0

Solving
-51 + 2n + n2 = 0

Solving for variable 'n'.

Begin completing the square.

Move the constant term to the right:

Add '51' to each side of the equation.
-51 + 2n + 51 + n2 = 0 + 51

Reorder the terms:
-51 + 51 + 2n + n2 = 0 + 51

Combine like terms: -51 + 51 = 0
0 + 2n + n2 = 0 + 51
2n + n2 = 0 + 51

Combine like terms: 0 + 51 = 51
2n + n2 = 51

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 = 51 + 1

Reorder the terms:
1 + 2n + n2 = 51 + 1

Combine like terms: 51 + 1 = 52
1 + 2n + n2 = 52

Factor a perfect square on the left side:
(n + 1)(n + 1) = 52

Calculate the square root of the right side: 7.211102551

Break this problem into two subproblems by setting 
(n + 1) equal to 7.211102551 and -7.211102551.

Subproblem 1

n + 1 = 7.211102551 Simplifying n + 1 = 7.211102551 Reorder the terms: 1 + n = 7.211102551 Solving 1 + n = 7.211102551 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 = 7.211102551 + -1 Combine like terms: 1 + -1 = 0 0 + n = 7.211102551 + -1 n = 7.211102551 + -1 Combine like terms: 7.211102551 + -1 = 6.211102551 n = 6.211102551 Simplifying n = 6.211102551

Subproblem 2

n + 1 = -7.211102551 Simplifying n + 1 = -7.211102551 Reorder the terms: 1 + n = -7.211102551 Solving 1 + n = -7.211102551 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 = -7.211102551 + -1 Combine like terms: 1 + -1 = 0 0 + n = -7.211102551 + -1 n = -7.211102551 + -1 Combine like terms: -7.211102551 + -1 = -8.211102551 n = -8.211102551 Simplifying n = -8.211102551

Solution

The solution to the problem is based on the solutions from the subproblems. n = {6.211102551, -8.211102551}

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