n^2-8n-52=9

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Solution for n^2-8n-52=9 equation:


Simplifying
n2 + -8n + -52 = 9

Reorder the terms:
-52 + -8n + n2 = 9

Solving
-52 + -8n + n2 = 9

Solving for variable 'n'.

Reorder the terms:
-52 + -9 + -8n + n2 = 9 + -9

Combine like terms: -52 + -9 = -61
-61 + -8n + n2 = 9 + -9

Combine like terms: 9 + -9 = 0
-61 + -8n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '61' to each side of the equation.
-61 + -8n + 61 + n2 = 0 + 61

Reorder the terms:
-61 + 61 + -8n + n2 = 0 + 61

Combine like terms: -61 + 61 = 0
0 + -8n + n2 = 0 + 61
-8n + n2 = 0 + 61

Combine like terms: 0 + 61 = 61
-8n + n2 = 61

The n term is -8n.  Take half its coefficient (-4).
Square it (16) and add it to both sides.

Add '16' to each side of the equation.
-8n + 16 + n2 = 61 + 16

Reorder the terms:
16 + -8n + n2 = 61 + 16

Combine like terms: 61 + 16 = 77
16 + -8n + n2 = 77

Factor a perfect square on the left side:
(n + -4)(n + -4) = 77

Calculate the square root of the right side: 8.774964387

Break this problem into two subproblems by setting 
(n + -4) equal to 8.774964387 and -8.774964387.

Subproblem 1

n + -4 = 8.774964387 Simplifying n + -4 = 8.774964387 Reorder the terms: -4 + n = 8.774964387 Solving -4 + n = 8.774964387 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + n = 8.774964387 + 4 Combine like terms: -4 + 4 = 0 0 + n = 8.774964387 + 4 n = 8.774964387 + 4 Combine like terms: 8.774964387 + 4 = 12.774964387 n = 12.774964387 Simplifying n = 12.774964387

Subproblem 2

n + -4 = -8.774964387 Simplifying n + -4 = -8.774964387 Reorder the terms: -4 + n = -8.774964387 Solving -4 + n = -8.774964387 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + n = -8.774964387 + 4 Combine like terms: -4 + 4 = 0 0 + n = -8.774964387 + 4 n = -8.774964387 + 4 Combine like terms: -8.774964387 + 4 = -4.774964387 n = -4.774964387 Simplifying n = -4.774964387

Solution

The solution to the problem is based on the solutions from the subproblems. n = {12.774964387, -4.774964387}

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