n^2+5n=12

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Solution for n^2+5n=12 equation:


Simplifying
n2 + 5n = 12

Reorder the terms:
5n + n2 = 12

Solving
5n + n2 = 12

Solving for variable 'n'.

Reorder the terms:
-12 + 5n + n2 = 12 + -12

Combine like terms: 12 + -12 = 0
-12 + 5n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '12' to each side of the equation.
-12 + 5n + 12 + n2 = 0 + 12

Reorder the terms:
-12 + 12 + 5n + n2 = 0 + 12

Combine like terms: -12 + 12 = 0
0 + 5n + n2 = 0 + 12
5n + n2 = 0 + 12

Combine like terms: 0 + 12 = 12
5n + n2 = 12

The n term is 5n.  Take half its coefficient (2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
5n + 6.25 + n2 = 12 + 6.25

Reorder the terms:
6.25 + 5n + n2 = 12 + 6.25

Combine like terms: 12 + 6.25 = 18.25
6.25 + 5n + n2 = 18.25

Factor a perfect square on the left side:
(n + 2.5)(n + 2.5) = 18.25

Calculate the square root of the right side: 4.272001873

Break this problem into two subproblems by setting 
(n + 2.5) equal to 4.272001873 and -4.272001873.

Subproblem 1

n + 2.5 = 4.272001873 Simplifying n + 2.5 = 4.272001873 Reorder the terms: 2.5 + n = 4.272001873 Solving 2.5 + n = 4.272001873 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + n = 4.272001873 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + n = 4.272001873 + -2.5 n = 4.272001873 + -2.5 Combine like terms: 4.272001873 + -2.5 = 1.772001873 n = 1.772001873 Simplifying n = 1.772001873

Subproblem 2

n + 2.5 = -4.272001873 Simplifying n + 2.5 = -4.272001873 Reorder the terms: 2.5 + n = -4.272001873 Solving 2.5 + n = -4.272001873 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + n = -4.272001873 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + n = -4.272001873 + -2.5 n = -4.272001873 + -2.5 Combine like terms: -4.272001873 + -2.5 = -6.772001873 n = -6.772001873 Simplifying n = -6.772001873

Solution

The solution to the problem is based on the solutions from the subproblems. n = {1.772001873, -6.772001873}

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