4n^2+2n=21

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Solution for 4n^2+2n=21 equation:


Simplifying
4n2 + 2n = 21

Reorder the terms:
2n + 4n2 = 21

Solving
2n + 4n2 = 21

Solving for variable 'n'.

Reorder the terms:
-21 + 2n + 4n2 = 21 + -21

Combine like terms: 21 + -21 = 0
-21 + 2n + 4n2 = 0

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

Divide each side by '4'.
-5.25 + 0.5n + n2 = 0

Move the constant term to the right:

Add '5.25' to each side of the equation.
-5.25 + 0.5n + 5.25 + n2 = 0 + 5.25

Reorder the terms:
-5.25 + 5.25 + 0.5n + n2 = 0 + 5.25

Combine like terms: -5.25 + 5.25 = 0.00
0.00 + 0.5n + n2 = 0 + 5.25
0.5n + n2 = 0 + 5.25

Combine like terms: 0 + 5.25 = 5.25
0.5n + n2 = 5.25

The n term is 0.5n.  Take half its coefficient (0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
0.5n + 0.0625 + n2 = 5.25 + 0.0625

Reorder the terms:
0.0625 + 0.5n + n2 = 5.25 + 0.0625

Combine like terms: 5.25 + 0.0625 = 5.3125
0.0625 + 0.5n + n2 = 5.3125

Factor a perfect square on the left side:
(n + 0.25)(n + 0.25) = 5.3125

Calculate the square root of the right side: 2.304886114

Break this problem into two subproblems by setting 
(n + 0.25) equal to 2.304886114 and -2.304886114.

Subproblem 1

n + 0.25 = 2.304886114 Simplifying n + 0.25 = 2.304886114 Reorder the terms: 0.25 + n = 2.304886114 Solving 0.25 + n = 2.304886114 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + n = 2.304886114 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + n = 2.304886114 + -0.25 n = 2.304886114 + -0.25 Combine like terms: 2.304886114 + -0.25 = 2.054886114 n = 2.054886114 Simplifying n = 2.054886114

Subproblem 2

n + 0.25 = -2.304886114 Simplifying n + 0.25 = -2.304886114 Reorder the terms: 0.25 + n = -2.304886114 Solving 0.25 + n = -2.304886114 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + n = -2.304886114 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + n = -2.304886114 + -0.25 n = -2.304886114 + -0.25 Combine like terms: -2.304886114 + -0.25 = -2.554886114 n = -2.554886114 Simplifying n = -2.554886114

Solution

The solution to the problem is based on the solutions from the subproblems. n = {2.054886114, -2.554886114}

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