n^2+3n-12=0

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Solution for n^2+3n-12=0 equation:


Simplifying
n2 + 3n + -12 = 0

Reorder the terms:
-12 + 3n + n2 = 0

Solving
-12 + 3n + n2 = 0

Solving for variable 'n'.

Begin completing the square.

Move the constant term to the right:

Add '12' to each side of the equation.
-12 + 3n + 12 + n2 = 0 + 12

Reorder the terms:
-12 + 12 + 3n + n2 = 0 + 12

Combine like terms: -12 + 12 = 0
0 + 3n + n2 = 0 + 12
3n + n2 = 0 + 12

Combine like terms: 0 + 12 = 12
3n + n2 = 12

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 = 12 + 2.25

Reorder the terms:
2.25 + 3n + n2 = 12 + 2.25

Combine like terms: 12 + 2.25 = 14.25
2.25 + 3n + n2 = 14.25

Factor a perfect square on the left side:
(n + 1.5)(n + 1.5) = 14.25

Calculate the square root of the right side: 3.774917218

Break this problem into two subproblems by setting 
(n + 1.5) equal to 3.774917218 and -3.774917218.

Subproblem 1

n + 1.5 = 3.774917218 Simplifying n + 1.5 = 3.774917218 Reorder the terms: 1.5 + n = 3.774917218 Solving 1.5 + n = 3.774917218 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 = 3.774917218 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = 3.774917218 + -1.5 n = 3.774917218 + -1.5 Combine like terms: 3.774917218 + -1.5 = 2.274917218 n = 2.274917218 Simplifying n = 2.274917218

Subproblem 2

n + 1.5 = -3.774917218 Simplifying n + 1.5 = -3.774917218 Reorder the terms: 1.5 + n = -3.774917218 Solving 1.5 + n = -3.774917218 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 = -3.774917218 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = -3.774917218 + -1.5 n = -3.774917218 + -1.5 Combine like terms: -3.774917218 + -1.5 = -5.274917218 n = -5.274917218 Simplifying n = -5.274917218

Solution

The solution to the problem is based on the solutions from the subproblems. n = {2.274917218, -5.274917218}

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