n^2-10n+13=0

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Solution for n^2-10n+13=0 equation:


Simplifying
n2 + -10n + 13 = 0

Reorder the terms:
13 + -10n + n2 = 0

Solving
13 + -10n + n2 = 0

Solving for variable 'n'.

Begin completing the square.

Move the constant term to the right:

Add '-13' to each side of the equation.
13 + -10n + -13 + n2 = 0 + -13

Reorder the terms:
13 + -13 + -10n + n2 = 0 + -13

Combine like terms: 13 + -13 = 0
0 + -10n + n2 = 0 + -13
-10n + n2 = 0 + -13

Combine like terms: 0 + -13 = -13
-10n + n2 = -13

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 = -13 + 25

Reorder the terms:
25 + -10n + n2 = -13 + 25

Combine like terms: -13 + 25 = 12
25 + -10n + n2 = 12

Factor a perfect square on the left side:
(n + -5)(n + -5) = 12

Calculate the square root of the right side: 3.464101615

Break this problem into two subproblems by setting 
(n + -5) equal to 3.464101615 and -3.464101615.

Subproblem 1

n + -5 = 3.464101615 Simplifying n + -5 = 3.464101615 Reorder the terms: -5 + n = 3.464101615 Solving -5 + n = 3.464101615 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 = 3.464101615 + 5 Combine like terms: -5 + 5 = 0 0 + n = 3.464101615 + 5 n = 3.464101615 + 5 Combine like terms: 3.464101615 + 5 = 8.464101615 n = 8.464101615 Simplifying n = 8.464101615

Subproblem 2

n + -5 = -3.464101615 Simplifying n + -5 = -3.464101615 Reorder the terms: -5 + n = -3.464101615 Solving -5 + n = -3.464101615 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 = -3.464101615 + 5 Combine like terms: -5 + 5 = 0 0 + n = -3.464101615 + 5 n = -3.464101615 + 5 Combine like terms: -3.464101615 + 5 = 1.535898385 n = 1.535898385 Simplifying n = 1.535898385

Solution

The solution to the problem is based on the solutions from the subproblems. n = {8.464101615, 1.535898385}

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