n^2-10n+13=-6

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Solution for n^2-10n+13=-6 equation:


Simplifying
n2 + -10n + 13 = -6

Reorder the terms:
13 + -10n + n2 = -6

Solving
13 + -10n + n2 = -6

Solving for variable 'n'.

Reorder the terms:
13 + 6 + -10n + n2 = -6 + 6

Combine like terms: 13 + 6 = 19
19 + -10n + n2 = -6 + 6

Combine like terms: -6 + 6 = 0
19 + -10n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-19' to each side of the equation.
19 + -10n + -19 + n2 = 0 + -19

Reorder the terms:
19 + -19 + -10n + n2 = 0 + -19

Combine like terms: 19 + -19 = 0
0 + -10n + n2 = 0 + -19
-10n + n2 = 0 + -19

Combine like terms: 0 + -19 = -19
-10n + n2 = -19

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 = -19 + 25

Reorder the terms:
25 + -10n + n2 = -19 + 25

Combine like terms: -19 + 25 = 6
25 + -10n + n2 = 6

Factor a perfect square on the left side:
(n + -5)(n + -5) = 6

Calculate the square root of the right side: 2.449489743

Break this problem into two subproblems by setting 
(n + -5) equal to 2.449489743 and -2.449489743.

Subproblem 1

n + -5 = 2.449489743 Simplifying n + -5 = 2.449489743 Reorder the terms: -5 + n = 2.449489743 Solving -5 + n = 2.449489743 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 = 2.449489743 + 5 Combine like terms: -5 + 5 = 0 0 + n = 2.449489743 + 5 n = 2.449489743 + 5 Combine like terms: 2.449489743 + 5 = 7.449489743 n = 7.449489743 Simplifying n = 7.449489743

Subproblem 2

n + -5 = -2.449489743 Simplifying n + -5 = -2.449489743 Reorder the terms: -5 + n = -2.449489743 Solving -5 + n = -2.449489743 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 = -2.449489743 + 5 Combine like terms: -5 + 5 = 0 0 + n = -2.449489743 + 5 n = -2.449489743 + 5 Combine like terms: -2.449489743 + 5 = 2.550510257 n = 2.550510257 Simplifying n = 2.550510257

Solution

The solution to the problem is based on the solutions from the subproblems. n = {7.449489743, 2.550510257}

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