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

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


Simplifying
4 + 9n + -3n2 = 2 + -1n2

Solving
4 + 9n + -3n2 = 2 + -1n2

Solving for variable 'n'.

Reorder the terms:
4 + -2 + 9n + -3n2 + n2 = 2 + -1n2 + -2 + n2

Combine like terms: 4 + -2 = 2
2 + 9n + -3n2 + n2 = 2 + -1n2 + -2 + n2

Combine like terms: -3n2 + n2 = -2n2
2 + 9n + -2n2 = 2 + -1n2 + -2 + n2

Reorder the terms:
2 + 9n + -2n2 = 2 + -2 + -1n2 + n2

Combine like terms: 2 + -2 = 0
2 + 9n + -2n2 = 0 + -1n2 + n2
2 + 9n + -2n2 = -1n2 + n2

Combine like terms: -1n2 + n2 = 0
2 + 9n + -2n2 = 0

Begin completing the square.  Divide all terms by
-2 the coefficient of the squared term: 

Divide each side by '-2'.
-1 + -4.5n + n2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + -4.5n + 1 + n2 = 0 + 1

Reorder the terms:
-1 + 1 + -4.5n + n2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + -4.5n + n2 = 0 + 1
-4.5n + n2 = 0 + 1

Combine like terms: 0 + 1 = 1
-4.5n + n2 = 1

The n term is -4.5n.  Take half its coefficient (-2.25).
Square it (5.0625) and add it to both sides.

Add '5.0625' to each side of the equation.
-4.5n + 5.0625 + n2 = 1 + 5.0625

Reorder the terms:
5.0625 + -4.5n + n2 = 1 + 5.0625

Combine like terms: 1 + 5.0625 = 6.0625
5.0625 + -4.5n + n2 = 6.0625

Factor a perfect square on the left side:
(n + -2.25)(n + -2.25) = 6.0625

Calculate the square root of the right side: 2.46221445

Break this problem into two subproblems by setting 
(n + -2.25) equal to 2.46221445 and -2.46221445.

Subproblem 1

n + -2.25 = 2.46221445 Simplifying n + -2.25 = 2.46221445 Reorder the terms: -2.25 + n = 2.46221445 Solving -2.25 + n = 2.46221445 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '2.25' to each side of the equation. -2.25 + 2.25 + n = 2.46221445 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + n = 2.46221445 + 2.25 n = 2.46221445 + 2.25 Combine like terms: 2.46221445 + 2.25 = 4.71221445 n = 4.71221445 Simplifying n = 4.71221445

Subproblem 2

n + -2.25 = -2.46221445 Simplifying n + -2.25 = -2.46221445 Reorder the terms: -2.25 + n = -2.46221445 Solving -2.25 + n = -2.46221445 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '2.25' to each side of the equation. -2.25 + 2.25 + n = -2.46221445 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + n = -2.46221445 + 2.25 n = -2.46221445 + 2.25 Combine like terms: -2.46221445 + 2.25 = -0.21221445 n = -0.21221445 Simplifying n = -0.21221445

Solution

The solution to the problem is based on the solutions from the subproblems. n = {4.71221445, -0.21221445}

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