0.5*n^2+n=6

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Solution for 0.5*n^2+n=6 equation:


Simplifying
0.5n2 + n = 6

Reorder the terms:
n + 0.5n2 = 6

Solving
n + 0.5n2 = 6

Solving for variable 'n'.

Reorder the terms:
-6 + n + 0.5n2 = 6 + -6

Combine like terms: 6 + -6 = 0
-6 + n + 0.5n2 = 0

Begin completing the square.  Divide all terms by
0.5 the coefficient of the squared term: 

Divide each side by '0.5'.
-12 + 2n + n2 = 0

Move the constant term to the right:

Add '12' to each side of the equation.
-12 + 2n + 12 + n2 = 0 + 12

Reorder the terms:
-12 + 12 + 2n + n2 = 0 + 12

Combine like terms: -12 + 12 = 0
0 + 2n + n2 = 0 + 12
2n + n2 = 0 + 12

Combine like terms: 0 + 12 = 12
2n + n2 = 12

The n term is n.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
2n + 0.25 + n2 = 12 + 0.25

Reorder the terms:
0.25 + 2n + n2 = 12 + 0.25

Combine like terms: 12 + 0.25 = 12.25
0.25 + 2n + n2 = 12.25

Factor a perfect square on the left side:
(n + 0.5)(n + 0.5) = 12.25

Calculate the square root of the right side: 3.5

Break this problem into two subproblems by setting 
(n + 0.5) equal to 3.5 and -3.5.

Subproblem 1

n + 0.5 = 3.5 Simplifying n + 0.5 = 3.5 Reorder the terms: 0.5 + n = 3.5 Solving 0.5 + n = 3.5 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = 3.5 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 3.5 + -0.5 n = 3.5 + -0.5 Combine like terms: 3.5 + -0.5 = 3 n = 3 Simplifying n = 3

Subproblem 2

n + 0.5 = -3.5 Simplifying n + 0.5 = -3.5 Reorder the terms: 0.5 + n = -3.5 Solving 0.5 + n = -3.5 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = -3.5 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -3.5 + -0.5 n = -3.5 + -0.5 Combine like terms: -3.5 + -0.5 = -4 n = -4 Simplifying n = -4

Solution

The solution to the problem is based on the solutions from the subproblems. n = {3, -4}

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