4+6n=9n^2

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


Simplifying
4 + 6n = 9n2

Solving
4 + 6n = 9n2

Solving for variable 'n'.

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

Begin completing the square.  Divide all terms by
-9 the coefficient of the squared term: 

Divide each side by '-9'.
-0.4444444444 + -0.6666666667n + n2 = 0

Move the constant term to the right:

Add '0.4444444444' to each side of the equation.
-0.4444444444 + -0.6666666667n + 0.4444444444 + n2 = 0 + 0.4444444444

Reorder the terms:
-0.4444444444 + 0.4444444444 + -0.6666666667n + n2 = 0 + 0.4444444444

Combine like terms: -0.4444444444 + 0.4444444444 = 0.0000000000
0.0000000000 + -0.6666666667n + n2 = 0 + 0.4444444444
-0.6666666667n + n2 = 0 + 0.4444444444

Combine like terms: 0 + 0.4444444444 = 0.4444444444
-0.6666666667n + n2 = 0.4444444444

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 = 0.4444444444 + 0.1111111112

Reorder the terms:
0.1111111112 + -0.6666666667n + n2 = 0.4444444444 + 0.1111111112

Combine like terms: 0.4444444444 + 0.1111111112 = 0.5555555556
0.1111111112 + -0.6666666667n + n2 = 0.5555555556

Factor a perfect square on the left side:
(n + -0.3333333334)(n + -0.3333333334) = 0.5555555556

Calculate the square root of the right side: 0.745355993

Break this problem into two subproblems by setting 
(n + -0.3333333334) equal to 0.745355993 and -0.745355993.

Subproblem 1

n + -0.3333333334 = 0.745355993 Simplifying n + -0.3333333334 = 0.745355993 Reorder the terms: -0.3333333334 + n = 0.745355993 Solving -0.3333333334 + n = 0.745355993 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 = 0.745355993 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + n = 0.745355993 + 0.3333333334 n = 0.745355993 + 0.3333333334 Combine like terms: 0.745355993 + 0.3333333334 = 1.0786893264 n = 1.0786893264 Simplifying n = 1.0786893264

Subproblem 2

n + -0.3333333334 = -0.745355993 Simplifying n + -0.3333333334 = -0.745355993 Reorder the terms: -0.3333333334 + n = -0.745355993 Solving -0.3333333334 + n = -0.745355993 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 = -0.745355993 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + n = -0.745355993 + 0.3333333334 n = -0.745355993 + 0.3333333334 Combine like terms: -0.745355993 + 0.3333333334 = -0.4120226596 n = -0.4120226596 Simplifying n = -0.4120226596

Solution

The solution to the problem is based on the solutions from the subproblems. n = {1.0786893264, -0.4120226596}

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