n^2+14n-78=-4

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Solution for n^2+14n-78=-4 equation:


Simplifying
n2 + 14n + -78 = -4

Reorder the terms:
-78 + 14n + n2 = -4

Solving
-78 + 14n + n2 = -4

Solving for variable 'n'.

Reorder the terms:
-78 + 4 + 14n + n2 = -4 + 4

Combine like terms: -78 + 4 = -74
-74 + 14n + n2 = -4 + 4

Combine like terms: -4 + 4 = 0
-74 + 14n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '74' to each side of the equation.
-74 + 14n + 74 + n2 = 0 + 74

Reorder the terms:
-74 + 74 + 14n + n2 = 0 + 74

Combine like terms: -74 + 74 = 0
0 + 14n + n2 = 0 + 74
14n + n2 = 0 + 74

Combine like terms: 0 + 74 = 74
14n + n2 = 74

The n term is 14n.  Take half its coefficient (7).
Square it (49) and add it to both sides.

Add '49' to each side of the equation.
14n + 49 + n2 = 74 + 49

Reorder the terms:
49 + 14n + n2 = 74 + 49

Combine like terms: 74 + 49 = 123
49 + 14n + n2 = 123

Factor a perfect square on the left side:
(n + 7)(n + 7) = 123

Calculate the square root of the right side: 11.090536506

Break this problem into two subproblems by setting 
(n + 7) equal to 11.090536506 and -11.090536506.

Subproblem 1

n + 7 = 11.090536506 Simplifying n + 7 = 11.090536506 Reorder the terms: 7 + n = 11.090536506 Solving 7 + n = 11.090536506 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-7' to each side of the equation. 7 + -7 + n = 11.090536506 + -7 Combine like terms: 7 + -7 = 0 0 + n = 11.090536506 + -7 n = 11.090536506 + -7 Combine like terms: 11.090536506 + -7 = 4.090536506 n = 4.090536506 Simplifying n = 4.090536506

Subproblem 2

n + 7 = -11.090536506 Simplifying n + 7 = -11.090536506 Reorder the terms: 7 + n = -11.090536506 Solving 7 + n = -11.090536506 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-7' to each side of the equation. 7 + -7 + n = -11.090536506 + -7 Combine like terms: 7 + -7 = 0 0 + n = -11.090536506 + -7 n = -11.090536506 + -7 Combine like terms: -11.090536506 + -7 = -18.090536506 n = -18.090536506 Simplifying n = -18.090536506

Solution

The solution to the problem is based on the solutions from the subproblems. n = {4.090536506, -18.090536506}

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