6n^2+4n=21

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Solution for 6n^2+4n=21 equation:


Simplifying
6n2 + 4n = 21

Reorder the terms:
4n + 6n2 = 21

Solving
4n + 6n2 = 21

Solving for variable 'n'.

Reorder the terms:
-21 + 4n + 6n2 = 21 + -21

Combine like terms: 21 + -21 = 0
-21 + 4n + 6n2 = 0

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

Divide each side by '6'.
-3.5 + 0.6666666667n + n2 = 0

Move the constant term to the right:

Add '3.5' to each side of the equation.
-3.5 + 0.6666666667n + 3.5 + n2 = 0 + 3.5

Reorder the terms:
-3.5 + 3.5 + 0.6666666667n + n2 = 0 + 3.5

Combine like terms: -3.5 + 3.5 = 0.0
0.0 + 0.6666666667n + n2 = 0 + 3.5
0.6666666667n + n2 = 0 + 3.5

Combine like terms: 0 + 3.5 = 3.5
0.6666666667n + n2 = 3.5

The n term is 0.6666666667n.  Take half its coefficient (0.3333333334).
Square it (0.1111111112) and add it to both sides.

Add '0.1111111112' to each side of the equation.
0.6666666667n + 0.1111111112 + n2 = 3.5 + 0.1111111112

Reorder the terms:
0.1111111112 + 0.6666666667n + n2 = 3.5 + 0.1111111112

Combine like terms: 3.5 + 0.1111111112 = 3.6111111112
0.1111111112 + 0.6666666667n + n2 = 3.6111111112

Factor a perfect square on the left side:
(n + 0.3333333334)(n + 0.3333333334) = 3.6111111112

Calculate the square root of the right side: 1.900292375

Break this problem into two subproblems by setting 
(n + 0.3333333334) equal to 1.900292375 and -1.900292375.

Subproblem 1

n + 0.3333333334 = 1.900292375 Simplifying n + 0.3333333334 = 1.900292375 Reorder the terms: 0.3333333334 + n = 1.900292375 Solving 0.3333333334 + n = 1.900292375 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.3333333334' to each side of the equation. 0.3333333334 + -0.3333333334 + n = 1.900292375 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + n = 1.900292375 + -0.3333333334 n = 1.900292375 + -0.3333333334 Combine like terms: 1.900292375 + -0.3333333334 = 1.5669590416 n = 1.5669590416 Simplifying n = 1.5669590416

Subproblem 2

n + 0.3333333334 = -1.900292375 Simplifying n + 0.3333333334 = -1.900292375 Reorder the terms: 0.3333333334 + n = -1.900292375 Solving 0.3333333334 + n = -1.900292375 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.3333333334' to each side of the equation. 0.3333333334 + -0.3333333334 + n = -1.900292375 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + n = -1.900292375 + -0.3333333334 n = -1.900292375 + -0.3333333334 Combine like terms: -1.900292375 + -0.3333333334 = -2.2336257084 n = -2.2336257084 Simplifying n = -2.2336257084

Solution

The solution to the problem is based on the solutions from the subproblems. n = {1.5669590416, -2.2336257084}

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