n^2+6n-75=-5

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Solution for n^2+6n-75=-5 equation:


Simplifying
n2 + 6n + -75 = -5

Reorder the terms:
-75 + 6n + n2 = -5

Solving
-75 + 6n + n2 = -5

Solving for variable 'n'.

Reorder the terms:
-75 + 5 + 6n + n2 = -5 + 5

Combine like terms: -75 + 5 = -70
-70 + 6n + n2 = -5 + 5

Combine like terms: -5 + 5 = 0
-70 + 6n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '70' to each side of the equation.
-70 + 6n + 70 + n2 = 0 + 70

Reorder the terms:
-70 + 70 + 6n + n2 = 0 + 70

Combine like terms: -70 + 70 = 0
0 + 6n + n2 = 0 + 70
6n + n2 = 0 + 70

Combine like terms: 0 + 70 = 70
6n + n2 = 70

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 = 70 + 9

Reorder the terms:
9 + 6n + n2 = 70 + 9

Combine like terms: 70 + 9 = 79
9 + 6n + n2 = 79

Factor a perfect square on the left side:
(n + 3)(n + 3) = 79

Calculate the square root of the right side: 8.888194417

Break this problem into two subproblems by setting 
(n + 3) equal to 8.888194417 and -8.888194417.

Subproblem 1

n + 3 = 8.888194417 Simplifying n + 3 = 8.888194417 Reorder the terms: 3 + n = 8.888194417 Solving 3 + n = 8.888194417 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 = 8.888194417 + -3 Combine like terms: 3 + -3 = 0 0 + n = 8.888194417 + -3 n = 8.888194417 + -3 Combine like terms: 8.888194417 + -3 = 5.888194417 n = 5.888194417 Simplifying n = 5.888194417

Subproblem 2

n + 3 = -8.888194417 Simplifying n + 3 = -8.888194417 Reorder the terms: 3 + n = -8.888194417 Solving 3 + n = -8.888194417 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 = -8.888194417 + -3 Combine like terms: 3 + -3 = 0 0 + n = -8.888194417 + -3 n = -8.888194417 + -3 Combine like terms: -8.888194417 + -3 = -11.888194417 n = -11.888194417 Simplifying n = -11.888194417

Solution

The solution to the problem is based on the solutions from the subproblems. n = {5.888194417, -11.888194417}

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