4n+8(97+4n)-n(n+4)=9

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Solution for 4n+8(97+4n)-n(n+4)=9 equation:


Simplifying
4n + 8(97 + 4n) + -1n(n + 4) = 9
4n + (97 * 8 + 4n * 8) + -1n(n + 4) = 9
4n + (776 + 32n) + -1n(n + 4) = 9

Reorder the terms:
4n + 776 + 32n + -1n(4 + n) = 9
4n + 776 + 32n + (4 * -1n + n * -1n) = 9
4n + 776 + 32n + (-4n + -1n2) = 9

Reorder the terms:
776 + 4n + 32n + -4n + -1n2 = 9

Combine like terms: 4n + 32n = 36n
776 + 36n + -4n + -1n2 = 9

Combine like terms: 36n + -4n = 32n
776 + 32n + -1n2 = 9

Solving
776 + 32n + -1n2 = 9

Solving for variable 'n'.

Reorder the terms:
776 + -9 + 32n + -1n2 = 9 + -9

Combine like terms: 776 + -9 = 767
767 + 32n + -1n2 = 9 + -9

Combine like terms: 9 + -9 = 0
767 + 32n + -1n2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-767 + -32n + n2 = 0

Move the constant term to the right:

Add '767' to each side of the equation.
-767 + -32n + 767 + n2 = 0 + 767

Reorder the terms:
-767 + 767 + -32n + n2 = 0 + 767

Combine like terms: -767 + 767 = 0
0 + -32n + n2 = 0 + 767
-32n + n2 = 0 + 767

Combine like terms: 0 + 767 = 767
-32n + n2 = 767

The n term is -32n.  Take half its coefficient (-16).
Square it (256) and add it to both sides.

Add '256' to each side of the equation.
-32n + 256 + n2 = 767 + 256

Reorder the terms:
256 + -32n + n2 = 767 + 256

Combine like terms: 767 + 256 = 1023
256 + -32n + n2 = 1023

Factor a perfect square on the left side:
(n + -16)(n + -16) = 1023

Calculate the square root of the right side: 31.984371183

Break this problem into two subproblems by setting 
(n + -16) equal to 31.984371183 and -31.984371183.

Subproblem 1

n + -16 = 31.984371183 Simplifying n + -16 = 31.984371183 Reorder the terms: -16 + n = 31.984371183 Solving -16 + n = 31.984371183 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '16' to each side of the equation. -16 + 16 + n = 31.984371183 + 16 Combine like terms: -16 + 16 = 0 0 + n = 31.984371183 + 16 n = 31.984371183 + 16 Combine like terms: 31.984371183 + 16 = 47.984371183 n = 47.984371183 Simplifying n = 47.984371183

Subproblem 2

n + -16 = -31.984371183 Simplifying n + -16 = -31.984371183 Reorder the terms: -16 + n = -31.984371183 Solving -16 + n = -31.984371183 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '16' to each side of the equation. -16 + 16 + n = -31.984371183 + 16 Combine like terms: -16 + 16 = 0 0 + n = -31.984371183 + 16 n = -31.984371183 + 16 Combine like terms: -31.984371183 + 16 = -15.984371183 n = -15.984371183 Simplifying n = -15.984371183

Solution

The solution to the problem is based on the solutions from the subproblems. n = {47.984371183, -15.984371183}

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