-78-6n+n^2=0

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Solution for -78-6n+n^2=0 equation:


Simplifying
-78 + -6n + n2 = 0

Solving
-78 + -6n + n2 = 0

Solving for variable 'n'.

Begin completing the square.

Move the constant term to the right:

Add '78' to each side of the equation.
-78 + -6n + 78 + n2 = 0 + 78

Reorder the terms:
-78 + 78 + -6n + n2 = 0 + 78

Combine like terms: -78 + 78 = 0
0 + -6n + n2 = 0 + 78
-6n + n2 = 0 + 78

Combine like terms: 0 + 78 = 78
-6n + n2 = 78

The n term is -6n.  Take half its coefficient (-3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
-6n + 9 + n2 = 78 + 9

Reorder the terms:
9 + -6n + n2 = 78 + 9

Combine like terms: 78 + 9 = 87
9 + -6n + n2 = 87

Factor a perfect square on the left side:
(n + -3)(n + -3) = 87

Calculate the square root of the right side: 9.327379053

Break this problem into two subproblems by setting 
(n + -3) equal to 9.327379053 and -9.327379053.

Subproblem 1

n + -3 = 9.327379053 Simplifying n + -3 = 9.327379053 Reorder the terms: -3 + n = 9.327379053 Solving -3 + n = 9.327379053 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + n = 9.327379053 + 3 Combine like terms: -3 + 3 = 0 0 + n = 9.327379053 + 3 n = 9.327379053 + 3 Combine like terms: 9.327379053 + 3 = 12.327379053 n = 12.327379053 Simplifying n = 12.327379053

Subproblem 2

n + -3 = -9.327379053 Simplifying n + -3 = -9.327379053 Reorder the terms: -3 + n = -9.327379053 Solving -3 + n = -9.327379053 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + n = -9.327379053 + 3 Combine like terms: -3 + 3 = 0 0 + n = -9.327379053 + 3 n = -9.327379053 + 3 Combine like terms: -9.327379053 + 3 = -6.327379053 n = -6.327379053 Simplifying n = -6.327379053

Solution

The solution to the problem is based on the solutions from the subproblems. n = {12.327379053, -6.327379053}

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