n^2+12n-33=-10

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Solution for n^2+12n-33=-10 equation:


Simplifying
n2 + 12n + -33 = -10

Reorder the terms:
-33 + 12n + n2 = -10

Solving
-33 + 12n + n2 = -10

Solving for variable 'n'.

Reorder the terms:
-33 + 10 + 12n + n2 = -10 + 10

Combine like terms: -33 + 10 = -23
-23 + 12n + n2 = -10 + 10

Combine like terms: -10 + 10 = 0
-23 + 12n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '23' to each side of the equation.
-23 + 12n + 23 + n2 = 0 + 23

Reorder the terms:
-23 + 23 + 12n + n2 = 0 + 23

Combine like terms: -23 + 23 = 0
0 + 12n + n2 = 0 + 23
12n + n2 = 0 + 23

Combine like terms: 0 + 23 = 23
12n + n2 = 23

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 = 23 + 36

Reorder the terms:
36 + 12n + n2 = 23 + 36

Combine like terms: 23 + 36 = 59
36 + 12n + n2 = 59

Factor a perfect square on the left side:
(n + 6)(n + 6) = 59

Calculate the square root of the right side: 7.681145748

Break this problem into two subproblems by setting 
(n + 6) equal to 7.681145748 and -7.681145748.

Subproblem 1

n + 6 = 7.681145748 Simplifying n + 6 = 7.681145748 Reorder the terms: 6 + n = 7.681145748 Solving 6 + n = 7.681145748 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 = 7.681145748 + -6 Combine like terms: 6 + -6 = 0 0 + n = 7.681145748 + -6 n = 7.681145748 + -6 Combine like terms: 7.681145748 + -6 = 1.681145748 n = 1.681145748 Simplifying n = 1.681145748

Subproblem 2

n + 6 = -7.681145748 Simplifying n + 6 = -7.681145748 Reorder the terms: 6 + n = -7.681145748 Solving 6 + n = -7.681145748 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 = -7.681145748 + -6 Combine like terms: 6 + -6 = 0 0 + n = -7.681145748 + -6 n = -7.681145748 + -6 Combine like terms: -7.681145748 + -6 = -13.681145748 n = -13.681145748 Simplifying n = -13.681145748

Solution

The solution to the problem is based on the solutions from the subproblems. n = {1.681145748, -13.681145748}

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