2n+29=n^2+9

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Solution for 2n+29=n^2+9 equation:


Simplifying
2n + 29 = n2 + 9

Reorder the terms:
29 + 2n = n2 + 9

Reorder the terms:
29 + 2n = 9 + n2

Solving
29 + 2n = 9 + n2

Solving for variable 'n'.

Reorder the terms:
29 + -9 + 2n + -1n2 = 9 + n2 + -9 + -1n2

Combine like terms: 29 + -9 = 20
20 + 2n + -1n2 = 9 + n2 + -9 + -1n2

Reorder the terms:
20 + 2n + -1n2 = 9 + -9 + n2 + -1n2

Combine like terms: 9 + -9 = 0
20 + 2n + -1n2 = 0 + n2 + -1n2
20 + 2n + -1n2 = n2 + -1n2

Combine like terms: n2 + -1n2 = 0
20 + 2n + -1n2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-20 + -2n + n2 = 0

Move the constant term to the right:

Add '20' to each side of the equation.
-20 + -2n + 20 + n2 = 0 + 20

Reorder the terms:
-20 + 20 + -2n + n2 = 0 + 20

Combine like terms: -20 + 20 = 0
0 + -2n + n2 = 0 + 20
-2n + n2 = 0 + 20

Combine like terms: 0 + 20 = 20
-2n + n2 = 20

The n term is -2n.  Take half its coefficient (-1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
-2n + 1 + n2 = 20 + 1

Reorder the terms:
1 + -2n + n2 = 20 + 1

Combine like terms: 20 + 1 = 21
1 + -2n + n2 = 21

Factor a perfect square on the left side:
(n + -1)(n + -1) = 21

Calculate the square root of the right side: 4.582575695

Break this problem into two subproblems by setting 
(n + -1) equal to 4.582575695 and -4.582575695.

Subproblem 1

n + -1 = 4.582575695 Simplifying n + -1 = 4.582575695 Reorder the terms: -1 + n = 4.582575695 Solving -1 + n = 4.582575695 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + n = 4.582575695 + 1 Combine like terms: -1 + 1 = 0 0 + n = 4.582575695 + 1 n = 4.582575695 + 1 Combine like terms: 4.582575695 + 1 = 5.582575695 n = 5.582575695 Simplifying n = 5.582575695

Subproblem 2

n + -1 = -4.582575695 Simplifying n + -1 = -4.582575695 Reorder the terms: -1 + n = -4.582575695 Solving -1 + n = -4.582575695 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + n = -4.582575695 + 1 Combine like terms: -1 + 1 = 0 0 + n = -4.582575695 + 1 n = -4.582575695 + 1 Combine like terms: -4.582575695 + 1 = -3.582575695 n = -3.582575695 Simplifying n = -3.582575695

Solution

The solution to the problem is based on the solutions from the subproblems. n = {5.582575695, -3.582575695}

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