4n^2+4n-11=0

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Solution for 4n^2+4n-11=0 equation:


Simplifying
4n2 + 4n + -11 = 0

Reorder the terms:
-11 + 4n + 4n2 = 0

Solving
-11 + 4n + 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'.
-2.75 + n + n2 = 0

Move the constant term to the right:

Add '2.75' to each side of the equation.
-2.75 + n + 2.75 + n2 = 0 + 2.75

Reorder the terms:
-2.75 + 2.75 + n + n2 = 0 + 2.75

Combine like terms: -2.75 + 2.75 = 0.00
0.00 + n + n2 = 0 + 2.75
n + n2 = 0 + 2.75

Combine like terms: 0 + 2.75 = 2.75
n + n2 = 2.75

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.
 + 0.25 + n2 = 2.75 + 0.25

Combine like terms:  + 0.25 = 1.25
1.25 + n2 = 2.75 + 0.25

Combine like terms: 2.75 + 0.25 = 3
1.25 + n2 = 3

Factor a perfect square on the left side:
(n + 0.5)(n + 0.5) = 3

Calculate the square root of the right side: 1.732050808

Break this problem into two subproblems by setting 
(n + 0.5) equal to 1.732050808 and -1.732050808.

Subproblem 1

n + 0.5 = 1.732050808 Simplifying n + 0.5 = 1.732050808 Reorder the terms: 0.5 + n = 1.732050808 Solving 0.5 + n = 1.732050808 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 = 1.732050808 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 1.732050808 + -0.5 n = 1.732050808 + -0.5 Combine like terms: 1.732050808 + -0.5 = 1.232050808 n = 1.232050808 Simplifying n = 1.232050808

Subproblem 2

n + 0.5 = -1.732050808 Simplifying n + 0.5 = -1.732050808 Reorder the terms: 0.5 + n = -1.732050808 Solving 0.5 + n = -1.732050808 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 = -1.732050808 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -1.732050808 + -0.5 n = -1.732050808 + -0.5 Combine like terms: -1.732050808 + -0.5 = -2.232050808 n = -2.232050808 Simplifying n = -2.232050808

Solution

The solution to the problem is based on the solutions from the subproblems. n = {1.232050808, -2.232050808}

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