9n+4+3n^2-0.75=180

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Solution for 9n+4+3n^2-0.75=180 equation:


Simplifying
9n + 4 + 3n2 + -0.75 = 180

Reorder the terms:
4 + -0.75 + 9n + 3n2 = 180

Combine like terms: 4 + -0.75 = 3.25
3.25 + 9n + 3n2 = 180

Solving
3.25 + 9n + 3n2 = 180

Solving for variable 'n'.

Reorder the terms:
3.25 + -180 + 9n + 3n2 = 180 + -180

Combine like terms: 3.25 + -180 = -176.75
-176.75 + 9n + 3n2 = 180 + -180

Combine like terms: 180 + -180 = 0
-176.75 + 9n + 3n2 = 0

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

Divide each side by '3'.
-58.91666667 + 3n + n2 = 0

Move the constant term to the right:

Add '58.91666667' to each side of the equation.
-58.91666667 + 3n + 58.91666667 + n2 = 0 + 58.91666667

Reorder the terms:
-58.91666667 + 58.91666667 + 3n + n2 = 0 + 58.91666667

Combine like terms: -58.91666667 + 58.91666667 = 0.00000000
0.00000000 + 3n + n2 = 0 + 58.91666667
3n + n2 = 0 + 58.91666667

Combine like terms: 0 + 58.91666667 = 58.91666667
3n + n2 = 58.91666667

The n term is 3n.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3n + 2.25 + n2 = 58.91666667 + 2.25

Reorder the terms:
2.25 + 3n + n2 = 58.91666667 + 2.25

Combine like terms: 58.91666667 + 2.25 = 61.16666667
2.25 + 3n + n2 = 61.16666667

Factor a perfect square on the left side:
(n + 1.5)(n + 1.5) = 61.16666667

Calculate the square root of the right side: 7.820912138

Break this problem into two subproblems by setting 
(n + 1.5) equal to 7.820912138 and -7.820912138.

Subproblem 1

n + 1.5 = 7.820912138 Simplifying n + 1.5 = 7.820912138 Reorder the terms: 1.5 + n = 7.820912138 Solving 1.5 + n = 7.820912138 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + n = 7.820912138 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = 7.820912138 + -1.5 n = 7.820912138 + -1.5 Combine like terms: 7.820912138 + -1.5 = 6.320912138 n = 6.320912138 Simplifying n = 6.320912138

Subproblem 2

n + 1.5 = -7.820912138 Simplifying n + 1.5 = -7.820912138 Reorder the terms: 1.5 + n = -7.820912138 Solving 1.5 + n = -7.820912138 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + n = -7.820912138 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = -7.820912138 + -1.5 n = -7.820912138 + -1.5 Combine like terms: -7.820912138 + -1.5 = -9.320912138 n = -9.320912138 Simplifying n = -9.320912138

Solution

The solution to the problem is based on the solutions from the subproblems. n = {6.320912138, -9.320912138}

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