15n-9-12n^2+2-18+19n^2=

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Solution for 15n-9-12n^2+2-18+19n^2= equation:


Simplifying
15n + -9 + -12n2 + 2 + -18 + 19n2 = 0

Reorder the terms:
-9 + 2 + -18 + 15n + -12n2 + 19n2 = 0

Combine like terms: -9 + 2 = -7
-7 + -18 + 15n + -12n2 + 19n2 = 0

Combine like terms: -7 + -18 = -25
-25 + 15n + -12n2 + 19n2 = 0

Combine like terms: -12n2 + 19n2 = 7n2
-25 + 15n + 7n2 = 0

Solving
-25 + 15n + 7n2 = 0

Solving for variable 'n'.

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

Divide each side by '7'.
-3.571428571 + 2.142857143n + n2 = 0

Move the constant term to the right:

Add '3.571428571' to each side of the equation.
-3.571428571 + 2.142857143n + 3.571428571 + n2 = 0 + 3.571428571

Reorder the terms:
-3.571428571 + 3.571428571 + 2.142857143n + n2 = 0 + 3.571428571

Combine like terms: -3.571428571 + 3.571428571 = 0.000000000
0.000000000 + 2.142857143n + n2 = 0 + 3.571428571
2.142857143n + n2 = 0 + 3.571428571

Combine like terms: 0 + 3.571428571 = 3.571428571
2.142857143n + n2 = 3.571428571

The n term is 2.142857143n.  Take half its coefficient (1.071428572).
Square it (1.147959185) and add it to both sides.

Add '1.147959185' to each side of the equation.
2.142857143n + 1.147959185 + n2 = 3.571428571 + 1.147959185

Reorder the terms:
1.147959185 + 2.142857143n + n2 = 3.571428571 + 1.147959185

Combine like terms: 3.571428571 + 1.147959185 = 4.719387756
1.147959185 + 2.142857143n + n2 = 4.719387756

Factor a perfect square on the left side:
(n + 1.071428572)(n + 1.071428572) = 4.719387756

Calculate the square root of the right side: 2.17241519

Break this problem into two subproblems by setting 
(n + 1.071428572) equal to 2.17241519 and -2.17241519.

Subproblem 1

n + 1.071428572 = 2.17241519 Simplifying n + 1.071428572 = 2.17241519 Reorder the terms: 1.071428572 + n = 2.17241519 Solving 1.071428572 + n = 2.17241519 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.071428572' to each side of the equation. 1.071428572 + -1.071428572 + n = 2.17241519 + -1.071428572 Combine like terms: 1.071428572 + -1.071428572 = 0.000000000 0.000000000 + n = 2.17241519 + -1.071428572 n = 2.17241519 + -1.071428572 Combine like terms: 2.17241519 + -1.071428572 = 1.100986618 n = 1.100986618 Simplifying n = 1.100986618

Subproblem 2

n + 1.071428572 = -2.17241519 Simplifying n + 1.071428572 = -2.17241519 Reorder the terms: 1.071428572 + n = -2.17241519 Solving 1.071428572 + n = -2.17241519 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.071428572' to each side of the equation. 1.071428572 + -1.071428572 + n = -2.17241519 + -1.071428572 Combine like terms: 1.071428572 + -1.071428572 = 0.000000000 0.000000000 + n = -2.17241519 + -1.071428572 n = -2.17241519 + -1.071428572 Combine like terms: -2.17241519 + -1.071428572 = -3.243843762 n = -3.243843762 Simplifying n = -3.243843762

Solution

The solution to the problem is based on the solutions from the subproblems. n = {1.100986618, -3.243843762}

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