n^2-12n-21=0

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Solution for n^2-12n-21=0 equation:


Simplifying
n2 + -12n + -21 = 0

Reorder the terms:
-21 + -12n + n2 = 0

Solving
-21 + -12n + n2 = 0

Solving for variable 'n'.

Begin completing the square.

Move the constant term to the right:

Add '21' to each side of the equation.
-21 + -12n + 21 + n2 = 0 + 21

Reorder the terms:
-21 + 21 + -12n + n2 = 0 + 21

Combine like terms: -21 + 21 = 0
0 + -12n + n2 = 0 + 21
-12n + n2 = 0 + 21

Combine like terms: 0 + 21 = 21
-12n + n2 = 21

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 = 21 + 36

Reorder the terms:
36 + -12n + n2 = 21 + 36

Combine like terms: 21 + 36 = 57
36 + -12n + n2 = 57

Factor a perfect square on the left side:
(n + -6)(n + -6) = 57

Calculate the square root of the right side: 7.549834435

Break this problem into two subproblems by setting 
(n + -6) equal to 7.549834435 and -7.549834435.

Subproblem 1

n + -6 = 7.549834435 Simplifying n + -6 = 7.549834435 Reorder the terms: -6 + n = 7.549834435 Solving -6 + n = 7.549834435 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 = 7.549834435 + 6 Combine like terms: -6 + 6 = 0 0 + n = 7.549834435 + 6 n = 7.549834435 + 6 Combine like terms: 7.549834435 + 6 = 13.549834435 n = 13.549834435 Simplifying n = 13.549834435

Subproblem 2

n + -6 = -7.549834435 Simplifying n + -6 = -7.549834435 Reorder the terms: -6 + n = -7.549834435 Solving -6 + n = -7.549834435 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 = -7.549834435 + 6 Combine like terms: -6 + 6 = 0 0 + n = -7.549834435 + 6 n = -7.549834435 + 6 Combine like terms: -7.549834435 + 6 = -1.549834435 n = -1.549834435 Simplifying n = -1.549834435

Solution

The solution to the problem is based on the solutions from the subproblems. n = {13.549834435, -1.549834435}

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