9n^2-13=-6n

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Solution for 9n^2-13=-6n equation:


Simplifying
9n2 + -13 = -6n

Reorder the terms:
-13 + 9n2 = -6n

Solving
-13 + 9n2 = -6n

Solving for variable 'n'.

Reorder the terms:
-13 + 6n + 9n2 = -6n + 6n

Combine like terms: -6n + 6n = 0
-13 + 6n + 9n2 = 0

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

Divide each side by '9'.
-1.444444444 + 0.6666666667n + n2 = 0

Move the constant term to the right:

Add '1.444444444' to each side of the equation.
-1.444444444 + 0.6666666667n + 1.444444444 + n2 = 0 + 1.444444444

Reorder the terms:
-1.444444444 + 1.444444444 + 0.6666666667n + n2 = 0 + 1.444444444

Combine like terms: -1.444444444 + 1.444444444 = 0.000000000
0.000000000 + 0.6666666667n + n2 = 0 + 1.444444444
0.6666666667n + n2 = 0 + 1.444444444

Combine like terms: 0 + 1.444444444 = 1.444444444
0.6666666667n + n2 = 1.444444444

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 = 1.444444444 + 0.1111111112

Reorder the terms:
0.1111111112 + 0.6666666667n + n2 = 1.444444444 + 0.1111111112

Combine like terms: 1.444444444 + 0.1111111112 = 1.5555555552
0.1111111112 + 0.6666666667n + n2 = 1.5555555552

Factor a perfect square on the left side:
(n + 0.3333333334)(n + 0.3333333334) = 1.5555555552

Calculate the square root of the right side: 1.247219129

Break this problem into two subproblems by setting 
(n + 0.3333333334) equal to 1.247219129 and -1.247219129.

Subproblem 1

n + 0.3333333334 = 1.247219129 Simplifying n + 0.3333333334 = 1.247219129 Reorder the terms: 0.3333333334 + n = 1.247219129 Solving 0.3333333334 + n = 1.247219129 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 = 1.247219129 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + n = 1.247219129 + -0.3333333334 n = 1.247219129 + -0.3333333334 Combine like terms: 1.247219129 + -0.3333333334 = 0.9138857956 n = 0.9138857956 Simplifying n = 0.9138857956

Subproblem 2

n + 0.3333333334 = -1.247219129 Simplifying n + 0.3333333334 = -1.247219129 Reorder the terms: 0.3333333334 + n = -1.247219129 Solving 0.3333333334 + n = -1.247219129 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 = -1.247219129 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + n = -1.247219129 + -0.3333333334 n = -1.247219129 + -0.3333333334 Combine like terms: -1.247219129 + -0.3333333334 = -1.5805524624 n = -1.5805524624 Simplifying n = -1.5805524624

Solution

The solution to the problem is based on the solutions from the subproblems. n = {0.9138857956, -1.5805524624}

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