3n^2-0.75=9n+4

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


Simplifying
3n2 + -0.75 = 9n + 4

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

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

Solving
-0.75 + 3n2 = 4 + 9n

Solving for variable 'n'.

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

Combine like terms: -0.75 + -4 = -4.75
-4.75 + -9n + 3n2 = 4 + 9n + -4 + -9n

Reorder the terms:
-4.75 + -9n + 3n2 = 4 + -4 + 9n + -9n

Combine like terms: 4 + -4 = 0
-4.75 + -9n + 3n2 = 0 + 9n + -9n
-4.75 + -9n + 3n2 = 9n + -9n

Combine like terms: 9n + -9n = 0
-4.75 + -9n + 3n2 = 0

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

Divide each side by '3'.
-1.583333333 + -3n + n2 = 0

Move the constant term to the right:

Add '1.583333333' to each side of the equation.
-1.583333333 + -3n + 1.583333333 + n2 = 0 + 1.583333333

Reorder the terms:
-1.583333333 + 1.583333333 + -3n + n2 = 0 + 1.583333333

Combine like terms: -1.583333333 + 1.583333333 = 0.000000000
0.000000000 + -3n + n2 = 0 + 1.583333333
-3n + n2 = 0 + 1.583333333

Combine like terms: 0 + 1.583333333 = 1.583333333
-3n + n2 = 1.583333333

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 = 1.583333333 + 2.25

Reorder the terms:
2.25 + -3n + n2 = 1.583333333 + 2.25

Combine like terms: 1.583333333 + 2.25 = 3.833333333
2.25 + -3n + n2 = 3.833333333

Factor a perfect square on the left side:
(n + -1.5)(n + -1.5) = 3.833333333

Calculate the square root of the right side: 1.957890021

Break this problem into two subproblems by setting 
(n + -1.5) equal to 1.957890021 and -1.957890021.

Subproblem 1

n + -1.5 = 1.957890021 Simplifying n + -1.5 = 1.957890021 Reorder the terms: -1.5 + n = 1.957890021 Solving -1.5 + n = 1.957890021 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 = 1.957890021 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = 1.957890021 + 1.5 n = 1.957890021 + 1.5 Combine like terms: 1.957890021 + 1.5 = 3.457890021 n = 3.457890021 Simplifying n = 3.457890021

Subproblem 2

n + -1.5 = -1.957890021 Simplifying n + -1.5 = -1.957890021 Reorder the terms: -1.5 + n = -1.957890021 Solving -1.5 + n = -1.957890021 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 = -1.957890021 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = -1.957890021 + 1.5 n = -1.957890021 + 1.5 Combine like terms: -1.957890021 + 1.5 = -0.457890021 n = -0.457890021 Simplifying n = -0.457890021

Solution

The solution to the problem is based on the solutions from the subproblems. n = {3.457890021, -0.457890021}

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