n^2-13n+22=7

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Solution for n^2-13n+22=7 equation:


Simplifying
n2 + -13n + 22 = 7

Reorder the terms:
22 + -13n + n2 = 7

Solving
22 + -13n + n2 = 7

Solving for variable 'n'.

Reorder the terms:
22 + -7 + -13n + n2 = 7 + -7

Combine like terms: 22 + -7 = 15
15 + -13n + n2 = 7 + -7

Combine like terms: 7 + -7 = 0
15 + -13n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-15' to each side of the equation.
15 + -13n + -15 + n2 = 0 + -15

Reorder the terms:
15 + -15 + -13n + n2 = 0 + -15

Combine like terms: 15 + -15 = 0
0 + -13n + n2 = 0 + -15
-13n + n2 = 0 + -15

Combine like terms: 0 + -15 = -15
-13n + n2 = -15

The n term is -13n.  Take half its coefficient (-6.5).
Square it (42.25) and add it to both sides.

Add '42.25' to each side of the equation.
-13n + 42.25 + n2 = -15 + 42.25

Reorder the terms:
42.25 + -13n + n2 = -15 + 42.25

Combine like terms: -15 + 42.25 = 27.25
42.25 + -13n + n2 = 27.25

Factor a perfect square on the left side:
(n + -6.5)(n + -6.5) = 27.25

Calculate the square root of the right side: 5.220153254

Break this problem into two subproblems by setting 
(n + -6.5) equal to 5.220153254 and -5.220153254.

Subproblem 1

n + -6.5 = 5.220153254 Simplifying n + -6.5 = 5.220153254 Reorder the terms: -6.5 + n = 5.220153254 Solving -6.5 + n = 5.220153254 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '6.5' to each side of the equation. -6.5 + 6.5 + n = 5.220153254 + 6.5 Combine like terms: -6.5 + 6.5 = 0.0 0.0 + n = 5.220153254 + 6.5 n = 5.220153254 + 6.5 Combine like terms: 5.220153254 + 6.5 = 11.720153254 n = 11.720153254 Simplifying n = 11.720153254

Subproblem 2

n + -6.5 = -5.220153254 Simplifying n + -6.5 = -5.220153254 Reorder the terms: -6.5 + n = -5.220153254 Solving -6.5 + n = -5.220153254 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '6.5' to each side of the equation. -6.5 + 6.5 + n = -5.220153254 + 6.5 Combine like terms: -6.5 + 6.5 = 0.0 0.0 + n = -5.220153254 + 6.5 n = -5.220153254 + 6.5 Combine like terms: -5.220153254 + 6.5 = 1.279846746 n = 1.279846746 Simplifying n = 1.279846746

Solution

The solution to the problem is based on the solutions from the subproblems. n = {11.720153254, 1.279846746}

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