4n^2+12n-3=0

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Solution for 4n^2+12n-3=0 equation:


Simplifying
4n2 + 12n + -3 = 0

Reorder the terms:
-3 + 12n + 4n2 = 0

Solving
-3 + 12n + 4n2 = 0

Solving for variable 'n'.

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 = 3
2.25 + 3n + n2 = 3

Factor a perfect square on the left side:
(n + 1.5)(n + 1.5) = 3

Calculate the square root of the right side: 1.732050808

Break this problem into two subproblems by setting 
(n + 1.5) equal to 1.732050808 and -1.732050808.

Subproblem 1

n + 1.5 = 1.732050808 Simplifying n + 1.5 = 1.732050808 Reorder the terms: 1.5 + n = 1.732050808 Solving 1.5 + n = 1.732050808 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 = 1.732050808 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = 1.732050808 + -1.5 n = 1.732050808 + -1.5 Combine like terms: 1.732050808 + -1.5 = 0.232050808 n = 0.232050808 Simplifying n = 0.232050808

Subproblem 2

n + 1.5 = -1.732050808 Simplifying n + 1.5 = -1.732050808 Reorder the terms: 1.5 + n = -1.732050808 Solving 1.5 + n = -1.732050808 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 = -1.732050808 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = -1.732050808 + -1.5 n = -1.732050808 + -1.5 Combine like terms: -1.732050808 + -1.5 = -3.232050808 n = -3.232050808 Simplifying n = -3.232050808

Solution

The solution to the problem is based on the solutions from the subproblems. n = {0.232050808, -3.232050808}

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