n^2-10n=21

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Solution for n^2-10n=21 equation:


Simplifying
n2 + -10n = 21

Reorder the terms:
-10n + n2 = 21

Solving
-10n + n2 = 21

Solving for variable 'n'.

Reorder the terms:
-21 + -10n + n2 = 21 + -21

Combine like terms: 21 + -21 = 0
-21 + -10n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '21' to each side of the equation.
-21 + -10n + 21 + n2 = 0 + 21

Reorder the terms:
-21 + 21 + -10n + n2 = 0 + 21

Combine like terms: -21 + 21 = 0
0 + -10n + n2 = 0 + 21
-10n + n2 = 0 + 21

Combine like terms: 0 + 21 = 21
-10n + n2 = 21

The n term is -10n.  Take half its coefficient (-5).
Square it (25) and add it to both sides.

Add '25' to each side of the equation.
-10n + 25 + n2 = 21 + 25

Reorder the terms:
25 + -10n + n2 = 21 + 25

Combine like terms: 21 + 25 = 46
25 + -10n + n2 = 46

Factor a perfect square on the left side:
(n + -5)(n + -5) = 46

Calculate the square root of the right side: 6.782329983

Break this problem into two subproblems by setting 
(n + -5) equal to 6.782329983 and -6.782329983.

Subproblem 1

n + -5 = 6.782329983 Simplifying n + -5 = 6.782329983 Reorder the terms: -5 + n = 6.782329983 Solving -5 + n = 6.782329983 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '5' to each side of the equation. -5 + 5 + n = 6.782329983 + 5 Combine like terms: -5 + 5 = 0 0 + n = 6.782329983 + 5 n = 6.782329983 + 5 Combine like terms: 6.782329983 + 5 = 11.782329983 n = 11.782329983 Simplifying n = 11.782329983

Subproblem 2

n + -5 = -6.782329983 Simplifying n + -5 = -6.782329983 Reorder the terms: -5 + n = -6.782329983 Solving -5 + n = -6.782329983 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '5' to each side of the equation. -5 + 5 + n = -6.782329983 + 5 Combine like terms: -5 + 5 = 0 0 + n = -6.782329983 + 5 n = -6.782329983 + 5 Combine like terms: -6.782329983 + 5 = -1.782329983 n = -1.782329983 Simplifying n = -1.782329983

Solution

The solution to the problem is based on the solutions from the subproblems. n = {11.782329983, -1.782329983}

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