v^2+22v+121=24

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Solution for v^2+22v+121=24 equation: 


Simplifying
v2 + 22v + 121 = 24

Reorder the terms:
121 + 22v + v2 = 24

Solving
121 + 22v + v2 = 24

Solving for variable 'v'.

Reorder the terms:
121 + -24 + 22v + v2 = 24 + -24

Combine like terms: 121 + -24 = 97
97 + 22v + v2 = 24 + -24

Combine like terms: 24 + -24 = 0
97 + 22v + v2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-97' to each side of the equation.
97 + 22v + -97 + v2 = 0 + -97

Reorder the terms:
97 + -97 + 22v + v2 = 0 + -97

Combine like terms: 97 + -97 = 0
0 + 22v + v2 = 0 + -97
22v + v2 = 0 + -97

Combine like terms: 0 + -97 = -97
22v + v2 = -97

The v term is 22v.  Take half its coefficient (11).
Square it (121) and add it to both sides.

Add '121' to each side of the equation.
22v + 121 + v2 = -97 + 121

Reorder the terms:
121 + 22v + v2 = -97 + 121

Combine like terms: -97 + 121 = 24
121 + 22v + v2 = 24

Factor a perfect square on the left side:
(v + 11)(v + 11) = 24

Calculate the square root of the right side: 4.898979486

Break this problem into two subproblems by setting 
(v + 11) equal to 4.898979486 and -4.898979486.


Subproblem 1
v + 11 = 4.898979486

Simplifying
v + 11 = 4.898979486

Reorder the terms:
11 + v = 4.898979486

Solving
11 + v = 4.898979486

Solving for variable 'v'.

Move all terms containing v to the left, all other terms to the right.

Add '-11' to each side of the equation.
11 + -11 + v = 4.898979486 + -11

Combine like terms: 11 + -11 = 0
0 + v = 4.898979486 + -11
v = 4.898979486 + -11

Combine like terms: 4.898979486 + -11 = -6.101020514
v = -6.101020514

Simplifying
v = -6.101020514


Subproblem 2
v + 11 = -4.898979486

Simplifying
v + 11 = -4.898979486

Reorder the terms:
11 + v = -4.898979486

Solving
11 + v = -4.898979486

Solving for variable 'v'.

Move all terms containing v to the left, all other terms to the right.

Add '-11' to each side of the equation.
11 + -11 + v = -4.898979486 + -11

Combine like terms: 11 + -11 = 0
0 + v = -4.898979486 + -11
v = -4.898979486 + -11

Combine like terms: -4.898979486 + -11 = -15.898979486
v = -15.898979486

Simplifying
v = -15.898979486


Solution
The solution to the problem is based on the solutions
from the subproblems.
v = {-6.101020514, -15.898979486}

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