n(n+4)+28=28(n-1)(n-8)

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Solution for n(n+4)+28=28(n-1)(n-8) equation:


Simplifying
n(n + 4) + 28 = 28(n + -1)(n + -8)

Reorder the terms:
n(4 + n) + 28 = 28(n + -1)(n + -8)
(4 * n + n * n) + 28 = 28(n + -1)(n + -8)
(4n + n2) + 28 = 28(n + -1)(n + -8)

Reorder the terms:
28 + 4n + n2 = 28(n + -1)(n + -8)

Reorder the terms:
28 + 4n + n2 = 28(-1 + n)(n + -8)

Reorder the terms:
28 + 4n + n2 = 28(-1 + n)(-8 + n)

Multiply (-1 + n) * (-8 + n)
28 + 4n + n2 = 28(-1(-8 + n) + n(-8 + n))
28 + 4n + n2 = 28((-8 * -1 + n * -1) + n(-8 + n))
28 + 4n + n2 = 28((8 + -1n) + n(-8 + n))
28 + 4n + n2 = 28(8 + -1n + (-8 * n + n * n))
28 + 4n + n2 = 28(8 + -1n + (-8n + n2))

Combine like terms: -1n + -8n = -9n
28 + 4n + n2 = 28(8 + -9n + n2)
28 + 4n + n2 = (8 * 28 + -9n * 28 + n2 * 28)
28 + 4n + n2 = (224 + -252n + 28n2)

Solving
28 + 4n + n2 = 224 + -252n + 28n2

Solving for variable 'n'.

Reorder the terms:
28 + -224 + 4n + 252n + n2 + -28n2 = 224 + -252n + 28n2 + -224 + 252n + -28n2

Combine like terms: 28 + -224 = -196
-196 + 4n + 252n + n2 + -28n2 = 224 + -252n + 28n2 + -224 + 252n + -28n2

Combine like terms: 4n + 252n = 256n
-196 + 256n + n2 + -28n2 = 224 + -252n + 28n2 + -224 + 252n + -28n2

Combine like terms: n2 + -28n2 = -27n2
-196 + 256n + -27n2 = 224 + -252n + 28n2 + -224 + 252n + -28n2

Reorder the terms:
-196 + 256n + -27n2 = 224 + -224 + -252n + 252n + 28n2 + -28n2

Combine like terms: 224 + -224 = 0
-196 + 256n + -27n2 = 0 + -252n + 252n + 28n2 + -28n2
-196 + 256n + -27n2 = -252n + 252n + 28n2 + -28n2

Combine like terms: -252n + 252n = 0
-196 + 256n + -27n2 = 0 + 28n2 + -28n2
-196 + 256n + -27n2 = 28n2 + -28n2

Combine like terms: 28n2 + -28n2 = 0
-196 + 256n + -27n2 = 0

Begin completing the square.  Divide all terms by
-27 the coefficient of the squared term: 

Divide each side by '-27'.
7.259259259 + -9.481481481n + n2 = 0

Move the constant term to the right:

Add '-7.259259259' to each side of the equation.
7.259259259 + -9.481481481n + -7.259259259 + n2 = 0 + -7.259259259

Reorder the terms:
7.259259259 + -7.259259259 + -9.481481481n + n2 = 0 + -7.259259259

Combine like terms: 7.259259259 + -7.259259259 = 0.000000000
0.000000000 + -9.481481481n + n2 = 0 + -7.259259259
-9.481481481n + n2 = 0 + -7.259259259

Combine like terms: 0 + -7.259259259 = -7.259259259
-9.481481481n + n2 = -7.259259259

The n term is -9.481481481n.  Take half its coefficient (-4.740740741).
Square it (22.47462277) and add it to both sides.

Add '22.47462277' to each side of the equation.
-9.481481481n + 22.47462277 + n2 = -7.259259259 + 22.47462277

Reorder the terms:
22.47462277 + -9.481481481n + n2 = -7.259259259 + 22.47462277

Combine like terms: -7.259259259 + 22.47462277 = 15.215363511
22.47462277 + -9.481481481n + n2 = 15.215363511

Factor a perfect square on the left side:
(n + -4.740740741)(n + -4.740740741) = 15.215363511

Calculate the square root of the right side: 3.900687569

Break this problem into two subproblems by setting 
(n + -4.740740741) equal to 3.900687569 and -3.900687569.

Subproblem 1

n + -4.740740741 = 3.900687569 Simplifying n + -4.740740741 = 3.900687569 Reorder the terms: -4.740740741 + n = 3.900687569 Solving -4.740740741 + n = 3.900687569 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '4.740740741' to each side of the equation. -4.740740741 + 4.740740741 + n = 3.900687569 + 4.740740741 Combine like terms: -4.740740741 + 4.740740741 = 0.000000000 0.000000000 + n = 3.900687569 + 4.740740741 n = 3.900687569 + 4.740740741 Combine like terms: 3.900687569 + 4.740740741 = 8.64142831 n = 8.64142831 Simplifying n = 8.64142831

Subproblem 2

n + -4.740740741 = -3.900687569 Simplifying n + -4.740740741 = -3.900687569 Reorder the terms: -4.740740741 + n = -3.900687569 Solving -4.740740741 + n = -3.900687569 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '4.740740741' to each side of the equation. -4.740740741 + 4.740740741 + n = -3.900687569 + 4.740740741 Combine like terms: -4.740740741 + 4.740740741 = 0.000000000 0.000000000 + n = -3.900687569 + 4.740740741 n = -3.900687569 + 4.740740741 Combine like terms: -3.900687569 + 4.740740741 = 0.840053172 n = 0.840053172 Simplifying n = 0.840053172

Solution

The solution to the problem is based on the solutions from the subproblems. n = {8.64142831, 0.840053172}

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