n^2+3n=6

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Solution for n^2+3n=6 equation:


Simplifying
n2 + 3n = 6

Reorder the terms:
3n + n2 = 6

Solving
3n + n2 = 6

Solving for variable 'n'.

Reorder the terms:
-6 + 3n + n2 = 6 + -6

Combine like terms: 6 + -6 = 0
-6 + 3n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '6' to each side of the equation.
-6 + 3n + 6 + n2 = 0 + 6

Reorder the terms:
-6 + 6 + 3n + n2 = 0 + 6

Combine like terms: -6 + 6 = 0
0 + 3n + n2 = 0 + 6
3n + n2 = 0 + 6

Combine like terms: 0 + 6 = 6
3n + n2 = 6

The n term is 3n.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3n + 2.25 + n2 = 6 + 2.25

Reorder the terms:
2.25 + 3n + n2 = 6 + 2.25

Combine like terms: 6 + 2.25 = 8.25
2.25 + 3n + n2 = 8.25

Factor a perfect square on the left side:
(n + 1.5)(n + 1.5) = 8.25

Calculate the square root of the right side: 2.872281323

Break this problem into two subproblems by setting 
(n + 1.5) equal to 2.872281323 and -2.872281323.

Subproblem 1

n + 1.5 = 2.872281323 Simplifying n + 1.5 = 2.872281323 Reorder the terms: 1.5 + n = 2.872281323 Solving 1.5 + n = 2.872281323 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + n = 2.872281323 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = 2.872281323 + -1.5 n = 2.872281323 + -1.5 Combine like terms: 2.872281323 + -1.5 = 1.372281323 n = 1.372281323 Simplifying n = 1.372281323

Subproblem 2

n + 1.5 = -2.872281323 Simplifying n + 1.5 = -2.872281323 Reorder the terms: 1.5 + n = -2.872281323 Solving 1.5 + n = -2.872281323 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + n = -2.872281323 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = -2.872281323 + -1.5 n = -2.872281323 + -1.5 Combine like terms: -2.872281323 + -1.5 = -4.372281323 n = -4.372281323 Simplifying n = -4.372281323

Solution

The solution to the problem is based on the solutions from the subproblems. n = {1.372281323, -4.372281323}

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