6n^2+4n=9

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Solution for 6n^2+4n=9 equation:


Simplifying
6n2 + 4n = 9

Reorder the terms:
4n + 6n2 = 9

Solving
4n + 6n2 = 9

Solving for variable 'n'.

Reorder the terms:
-9 + 4n + 6n2 = 9 + -9

Combine like terms: 9 + -9 = 0
-9 + 4n + 6n2 = 0

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

Divide each side by '6'.
-1.5 + 0.6666666667n + n2 = 0

Move the constant term to the right:

Add '1.5' to each side of the equation.
-1.5 + 0.6666666667n + 1.5 + n2 = 0 + 1.5

Reorder the terms:
-1.5 + 1.5 + 0.6666666667n + n2 = 0 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + 0.6666666667n + n2 = 0 + 1.5
0.6666666667n + n2 = 0 + 1.5

Combine like terms: 0 + 1.5 = 1.5
0.6666666667n + n2 = 1.5

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 = 1.5 + 0.1111111112

Reorder the terms:
0.1111111112 + 0.6666666667n + n2 = 1.5 + 0.1111111112

Combine like terms: 1.5 + 0.1111111112 = 1.6111111112
0.1111111112 + 0.6666666667n + n2 = 1.6111111112

Factor a perfect square on the left side:
(n + 0.3333333334)(n + 0.3333333334) = 1.6111111112

Calculate the square root of the right side: 1.269295518

Break this problem into two subproblems by setting 
(n + 0.3333333334) equal to 1.269295518 and -1.269295518.

Subproblem 1

n + 0.3333333334 = 1.269295518 Simplifying n + 0.3333333334 = 1.269295518 Reorder the terms: 0.3333333334 + n = 1.269295518 Solving 0.3333333334 + n = 1.269295518 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 = 1.269295518 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + n = 1.269295518 + -0.3333333334 n = 1.269295518 + -0.3333333334 Combine like terms: 1.269295518 + -0.3333333334 = 0.9359621846 n = 0.9359621846 Simplifying n = 0.9359621846

Subproblem 2

n + 0.3333333334 = -1.269295518 Simplifying n + 0.3333333334 = -1.269295518 Reorder the terms: 0.3333333334 + n = -1.269295518 Solving 0.3333333334 + n = -1.269295518 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 = -1.269295518 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + n = -1.269295518 + -0.3333333334 n = -1.269295518 + -0.3333333334 Combine like terms: -1.269295518 + -0.3333333334 = -1.6026288514 n = -1.6026288514 Simplifying n = -1.6026288514

Solution

The solution to the problem is based on the solutions from the subproblems. n = {0.9359621846, -1.6026288514}

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