4n^2=-12n-3

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Solution for 4n^2=-12n-3 equation:


Simplifying
4n2 = -12n + -3

Reorder the terms:
4n2 = -3 + -12n

Solving
4n2 = -3 + -12n

Solving for variable 'n'.

Reorder the terms:
3 + 12n + 4n2 = -3 + -12n + 3 + 12n

Reorder the terms:
3 + 12n + 4n2 = -3 + 3 + -12n + 12n

Combine like terms: -3 + 3 = 0
3 + 12n + 4n2 = 0 + -12n + 12n
3 + 12n + 4n2 = -12n + 12n

Combine like terms: -12n + 12n = 0
3 + 12n + 4n2 = 0

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

Divide each side by '4'.
0.75 + 3n + n2 = 0

Move the constant term to the right:

Add '-0.75' to each side of the equation.
0.75 + 3n + -0.75 + n2 = 0 + -0.75

Reorder the terms:
0.75 + -0.75 + 3n + n2 = 0 + -0.75

Combine like terms: 0.75 + -0.75 = 0.00
0.00 + 3n + n2 = 0 + -0.75
3n + n2 = 0 + -0.75

Combine like terms: 0 + -0.75 = -0.75
3n + n2 = -0.75

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 = -0.75 + 2.25

Reorder the terms:
2.25 + 3n + n2 = -0.75 + 2.25

Combine like terms: -0.75 + 2.25 = 1.5
2.25 + 3n + n2 = 1.5

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

Calculate the square root of the right side: 1.224744871

Break this problem into two subproblems by setting 
(n + 1.5) equal to 1.224744871 and -1.224744871.

Subproblem 1

n + 1.5 = 1.224744871 Simplifying n + 1.5 = 1.224744871 Reorder the terms: 1.5 + n = 1.224744871 Solving 1.5 + n = 1.224744871 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 = 1.224744871 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = 1.224744871 + -1.5 n = 1.224744871 + -1.5 Combine like terms: 1.224744871 + -1.5 = -0.275255129 n = -0.275255129 Simplifying n = -0.275255129

Subproblem 2

n + 1.5 = -1.224744871 Simplifying n + 1.5 = -1.224744871 Reorder the terms: 1.5 + n = -1.224744871 Solving 1.5 + n = -1.224744871 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 = -1.224744871 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = -1.224744871 + -1.5 n = -1.224744871 + -1.5 Combine like terms: -1.224744871 + -1.5 = -2.724744871 n = -2.724744871 Simplifying n = -2.724744871

Solution

The solution to the problem is based on the solutions from the subproblems. n = {-0.275255129, -2.724744871}

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