4n^2+4n+1=20

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Solution for 4n^2+4n+1=20 equation:


Simplifying
4n2 + 4n + 1 = 20

Reorder the terms:
1 + 4n + 4n2 = 20

Solving
1 + 4n + 4n2 = 20

Solving for variable 'n'.

Reorder the terms:
1 + -20 + 4n + 4n2 = 20 + -20

Combine like terms: 1 + -20 = -19
-19 + 4n + 4n2 = 20 + -20

Combine like terms: 20 + -20 = 0
-19 + 4n + 4n2 = 0

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

Divide each side by '4'.
-4.75 + n + n2 = 0

Move the constant term to the right:

Add '4.75' to each side of the equation.
-4.75 + n + 4.75 + n2 = 0 + 4.75

Reorder the terms:
-4.75 + 4.75 + n + n2 = 0 + 4.75

Combine like terms: -4.75 + 4.75 = 0.00
0.00 + n + n2 = 0 + 4.75
n + n2 = 0 + 4.75

Combine like terms: 0 + 4.75 = 4.75
n + n2 = 4.75

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 = 4.75 + 0.25

Combine like terms:  + 0.25 = 1.25
1.25 + n2 = 4.75 + 0.25

Combine like terms: 4.75 + 0.25 = 5
1.25 + n2 = 5

Factor a perfect square on the left side:
(n + 0.5)(n + 0.5) = 5

Calculate the square root of the right side: 2.236067978

Break this problem into two subproblems by setting 
(n + 0.5) equal to 2.236067978 and -2.236067978.

Subproblem 1

n + 0.5 = 2.236067978 Simplifying n + 0.5 = 2.236067978 Reorder the terms: 0.5 + n = 2.236067978 Solving 0.5 + n = 2.236067978 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 = 2.236067978 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 2.236067978 + -0.5 n = 2.236067978 + -0.5 Combine like terms: 2.236067978 + -0.5 = 1.736067978 n = 1.736067978 Simplifying n = 1.736067978

Subproblem 2

n + 0.5 = -2.236067978 Simplifying n + 0.5 = -2.236067978 Reorder the terms: 0.5 + n = -2.236067978 Solving 0.5 + n = -2.236067978 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 = -2.236067978 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -2.236067978 + -0.5 n = -2.236067978 + -0.5 Combine like terms: -2.236067978 + -0.5 = -2.736067978 n = -2.736067978 Simplifying n = -2.736067978

Solution

The solution to the problem is based on the solutions from the subproblems. n = {1.736067978, -2.736067978}

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