n^2+11n+29=21

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Solution for n^2+11n+29=21 equation:


Simplifying
n2 + 11n + 29 = 21

Reorder the terms:
29 + 11n + n2 = 21

Solving
29 + 11n + n2 = 21

Solving for variable 'n'.

Reorder the terms:
29 + -21 + 11n + n2 = 21 + -21

Combine like terms: 29 + -21 = 8
8 + 11n + n2 = 21 + -21

Combine like terms: 21 + -21 = 0
8 + 11n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-8' to each side of the equation.
8 + 11n + -8 + n2 = 0 + -8

Reorder the terms:
8 + -8 + 11n + n2 = 0 + -8

Combine like terms: 8 + -8 = 0
0 + 11n + n2 = 0 + -8
11n + n2 = 0 + -8

Combine like terms: 0 + -8 = -8
11n + n2 = -8

The n term is 11n.  Take half its coefficient (5.5).
Square it (30.25) and add it to both sides.

Add '30.25' to each side of the equation.
11n + 30.25 + n2 = -8 + 30.25

Reorder the terms:
30.25 + 11n + n2 = -8 + 30.25

Combine like terms: -8 + 30.25 = 22.25
30.25 + 11n + n2 = 22.25

Factor a perfect square on the left side:
(n + 5.5)(n + 5.5) = 22.25

Calculate the square root of the right side: 4.716990566

Break this problem into two subproblems by setting 
(n + 5.5) equal to 4.716990566 and -4.716990566.

Subproblem 1

n + 5.5 = 4.716990566 Simplifying n + 5.5 = 4.716990566 Reorder the terms: 5.5 + n = 4.716990566 Solving 5.5 + n = 4.716990566 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-5.5' to each side of the equation. 5.5 + -5.5 + n = 4.716990566 + -5.5 Combine like terms: 5.5 + -5.5 = 0.0 0.0 + n = 4.716990566 + -5.5 n = 4.716990566 + -5.5 Combine like terms: 4.716990566 + -5.5 = -0.783009434 n = -0.783009434 Simplifying n = -0.783009434

Subproblem 2

n + 5.5 = -4.716990566 Simplifying n + 5.5 = -4.716990566 Reorder the terms: 5.5 + n = -4.716990566 Solving 5.5 + n = -4.716990566 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-5.5' to each side of the equation. 5.5 + -5.5 + n = -4.716990566 + -5.5 Combine like terms: 5.5 + -5.5 = 0.0 0.0 + n = -4.716990566 + -5.5 n = -4.716990566 + -5.5 Combine like terms: -4.716990566 + -5.5 = -10.216990566 n = -10.216990566 Simplifying n = -10.216990566

Solution

The solution to the problem is based on the solutions from the subproblems. n = {-0.783009434, -10.216990566}

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