3n^2-21n-128=-5n-4n^2

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Solution for 3n^2-21n-128=-5n-4n^2 equation:


Simplifying
3n2 + -21n + -128 = -5n + -4n2

Reorder the terms:
-128 + -21n + 3n2 = -5n + -4n2

Solving
-128 + -21n + 3n2 = -5n + -4n2

Solving for variable 'n'.

Reorder the terms:
-128 + -21n + 5n + 3n2 + 4n2 = -5n + -4n2 + 5n + 4n2

Combine like terms: -21n + 5n = -16n
-128 + -16n + 3n2 + 4n2 = -5n + -4n2 + 5n + 4n2

Combine like terms: 3n2 + 4n2 = 7n2
-128 + -16n + 7n2 = -5n + -4n2 + 5n + 4n2

Reorder the terms:
-128 + -16n + 7n2 = -5n + 5n + -4n2 + 4n2

Combine like terms: -5n + 5n = 0
-128 + -16n + 7n2 = 0 + -4n2 + 4n2
-128 + -16n + 7n2 = -4n2 + 4n2

Combine like terms: -4n2 + 4n2 = 0
-128 + -16n + 7n2 = 0

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

Divide each side by '7'.
-18.28571429 + -2.285714286n + n2 = 0

Move the constant term to the right:

Add '18.28571429' to each side of the equation.
-18.28571429 + -2.285714286n + 18.28571429 + n2 = 0 + 18.28571429

Reorder the terms:
-18.28571429 + 18.28571429 + -2.285714286n + n2 = 0 + 18.28571429

Combine like terms: -18.28571429 + 18.28571429 = 0.00000000
0.00000000 + -2.285714286n + n2 = 0 + 18.28571429
-2.285714286n + n2 = 0 + 18.28571429

Combine like terms: 0 + 18.28571429 = 18.28571429
-2.285714286n + n2 = 18.28571429

The n term is -2.285714286n.  Take half its coefficient (-1.142857143).
Square it (1.306122449) and add it to both sides.

Add '1.306122449' to each side of the equation.
-2.285714286n + 1.306122449 + n2 = 18.28571429 + 1.306122449

Reorder the terms:
1.306122449 + -2.285714286n + n2 = 18.28571429 + 1.306122449

Combine like terms: 18.28571429 + 1.306122449 = 19.591836739
1.306122449 + -2.285714286n + n2 = 19.591836739

Factor a perfect square on the left side:
(n + -1.142857143)(n + -1.142857143) = 19.591836739

Calculate the square root of the right side: 4.426266682

Break this problem into two subproblems by setting 
(n + -1.142857143) equal to 4.426266682 and -4.426266682.

Subproblem 1

n + -1.142857143 = 4.426266682 Simplifying n + -1.142857143 = 4.426266682 Reorder the terms: -1.142857143 + n = 4.426266682 Solving -1.142857143 + n = 4.426266682 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.142857143' to each side of the equation. -1.142857143 + 1.142857143 + n = 4.426266682 + 1.142857143 Combine like terms: -1.142857143 + 1.142857143 = 0.000000000 0.000000000 + n = 4.426266682 + 1.142857143 n = 4.426266682 + 1.142857143 Combine like terms: 4.426266682 + 1.142857143 = 5.569123825 n = 5.569123825 Simplifying n = 5.569123825

Subproblem 2

n + -1.142857143 = -4.426266682 Simplifying n + -1.142857143 = -4.426266682 Reorder the terms: -1.142857143 + n = -4.426266682 Solving -1.142857143 + n = -4.426266682 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.142857143' to each side of the equation. -1.142857143 + 1.142857143 + n = -4.426266682 + 1.142857143 Combine like terms: -1.142857143 + 1.142857143 = 0.000000000 0.000000000 + n = -4.426266682 + 1.142857143 n = -4.426266682 + 1.142857143 Combine like terms: -4.426266682 + 1.142857143 = -3.283409539 n = -3.283409539 Simplifying n = -3.283409539

Solution

The solution to the problem is based on the solutions from the subproblems. n = {5.569123825, -3.283409539}

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