-4n^2+6n-16=-15n^2

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Solution for -4n^2+6n-16=-15n^2 equation:


Simplifying
-4n2 + 6n + -16 = -15n2

Reorder the terms:
-16 + 6n + -4n2 = -15n2

Solving
-16 + 6n + -4n2 = -15n2

Solving for variable 'n'.

Combine like terms: -4n2 + 15n2 = 11n2
-16 + 6n + 11n2 = -15n2 + 15n2

Combine like terms: -15n2 + 15n2 = 0
-16 + 6n + 11n2 = 0

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

Divide each side by '11'.
-1.454545455 + 0.5454545455n + n2 = 0

Move the constant term to the right:

Add '1.454545455' to each side of the equation.
-1.454545455 + 0.5454545455n + 1.454545455 + n2 = 0 + 1.454545455

Reorder the terms:
-1.454545455 + 1.454545455 + 0.5454545455n + n2 = 0 + 1.454545455

Combine like terms: -1.454545455 + 1.454545455 = 0.000000000
0.000000000 + 0.5454545455n + n2 = 0 + 1.454545455
0.5454545455n + n2 = 0 + 1.454545455

Combine like terms: 0 + 1.454545455 = 1.454545455
0.5454545455n + n2 = 1.454545455

The n term is 0.5454545455n.  Take half its coefficient (0.2727272728).
Square it (0.07438016533) and add it to both sides.

Add '0.07438016533' to each side of the equation.
0.5454545455n + 0.07438016533 + n2 = 1.454545455 + 0.07438016533

Reorder the terms:
0.07438016533 + 0.5454545455n + n2 = 1.454545455 + 0.07438016533

Combine like terms: 1.454545455 + 0.07438016533 = 1.52892562033
0.07438016533 + 0.5454545455n + n2 = 1.52892562033

Factor a perfect square on the left side:
(n + 0.2727272728)(n + 0.2727272728) = 1.52892562033

Calculate the square root of the right side: 1.236497319

Break this problem into two subproblems by setting 
(n + 0.2727272728) equal to 1.236497319 and -1.236497319.

Subproblem 1

n + 0.2727272728 = 1.236497319 Simplifying n + 0.2727272728 = 1.236497319 Reorder the terms: 0.2727272728 + n = 1.236497319 Solving 0.2727272728 + n = 1.236497319 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.2727272728' to each side of the equation. 0.2727272728 + -0.2727272728 + n = 1.236497319 + -0.2727272728 Combine like terms: 0.2727272728 + -0.2727272728 = 0.0000000000 0.0000000000 + n = 1.236497319 + -0.2727272728 n = 1.236497319 + -0.2727272728 Combine like terms: 1.236497319 + -0.2727272728 = 0.9637700462 n = 0.9637700462 Simplifying n = 0.9637700462

Subproblem 2

n + 0.2727272728 = -1.236497319 Simplifying n + 0.2727272728 = -1.236497319 Reorder the terms: 0.2727272728 + n = -1.236497319 Solving 0.2727272728 + n = -1.236497319 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.2727272728' to each side of the equation. 0.2727272728 + -0.2727272728 + n = -1.236497319 + -0.2727272728 Combine like terms: 0.2727272728 + -0.2727272728 = 0.0000000000 0.0000000000 + n = -1.236497319 + -0.2727272728 n = -1.236497319 + -0.2727272728 Combine like terms: -1.236497319 + -0.2727272728 = -1.5092245918 n = -1.5092245918 Simplifying n = -1.5092245918

Solution

The solution to the problem is based on the solutions from the subproblems. n = {0.9637700462, -1.5092245918}

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