n^2-12n-78=-9

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Solution for n^2-12n-78=-9 equation:


Simplifying
n2 + -12n + -78 = -9

Reorder the terms:
-78 + -12n + n2 = -9

Solving
-78 + -12n + n2 = -9

Solving for variable 'n'.

Reorder the terms:
-78 + 9 + -12n + n2 = -9 + 9

Combine like terms: -78 + 9 = -69
-69 + -12n + n2 = -9 + 9

Combine like terms: -9 + 9 = 0
-69 + -12n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '69' to each side of the equation.
-69 + -12n + 69 + n2 = 0 + 69

Reorder the terms:
-69 + 69 + -12n + n2 = 0 + 69

Combine like terms: -69 + 69 = 0
0 + -12n + n2 = 0 + 69
-12n + n2 = 0 + 69

Combine like terms: 0 + 69 = 69
-12n + n2 = 69

The n term is -12n.  Take half its coefficient (-6).
Square it (36) and add it to both sides.

Add '36' to each side of the equation.
-12n + 36 + n2 = 69 + 36

Reorder the terms:
36 + -12n + n2 = 69 + 36

Combine like terms: 69 + 36 = 105
36 + -12n + n2 = 105

Factor a perfect square on the left side:
(n + -6)(n + -6) = 105

Calculate the square root of the right side: 10.246950766

Break this problem into two subproblems by setting 
(n + -6) equal to 10.246950766 and -10.246950766.

Subproblem 1

n + -6 = 10.246950766 Simplifying n + -6 = 10.246950766 Reorder the terms: -6 + n = 10.246950766 Solving -6 + n = 10.246950766 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '6' to each side of the equation. -6 + 6 + n = 10.246950766 + 6 Combine like terms: -6 + 6 = 0 0 + n = 10.246950766 + 6 n = 10.246950766 + 6 Combine like terms: 10.246950766 + 6 = 16.246950766 n = 16.246950766 Simplifying n = 16.246950766

Subproblem 2

n + -6 = -10.246950766 Simplifying n + -6 = -10.246950766 Reorder the terms: -6 + n = -10.246950766 Solving -6 + n = -10.246950766 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '6' to each side of the equation. -6 + 6 + n = -10.246950766 + 6 Combine like terms: -6 + 6 = 0 0 + n = -10.246950766 + 6 n = -10.246950766 + 6 Combine like terms: -10.246950766 + 6 = -4.246950766 n = -4.246950766 Simplifying n = -4.246950766

Solution

The solution to the problem is based on the solutions from the subproblems. n = {16.246950766, -4.246950766}

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