2v^2+12v+25=v^2+1

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Solution for 2v^2+12v+25=v^2+1 equation:


Simplifying
2v2 + 12v + 25 = v2 + 1

Reorder the terms:
25 + 12v + 2v2 = v2 + 1

Reorder the terms:
25 + 12v + 2v2 = 1 + v2

Solving
25 + 12v + 2v2 = 1 + v2

Solving for variable 'v'.

Reorder the terms:
25 + -1 + 12v + 2v2 + -1v2 = 1 + v2 + -1 + -1v2

Combine like terms: 25 + -1 = 24
24 + 12v + 2v2 + -1v2 = 1 + v2 + -1 + -1v2

Combine like terms: 2v2 + -1v2 = 1v2
24 + 12v + 1v2 = 1 + v2 + -1 + -1v2

Reorder the terms:
24 + 12v + 1v2 = 1 + -1 + v2 + -1v2

Combine like terms: 1 + -1 = 0
24 + 12v + 1v2 = 0 + v2 + -1v2
24 + 12v + 1v2 = v2 + -1v2

Combine like terms: v2 + -1v2 = 0
24 + 12v + 1v2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-24' to each side of the equation.
24 + 12v + -24 + v2 = 0 + -24

Reorder the terms:
24 + -24 + 12v + v2 = 0 + -24

Combine like terms: 24 + -24 = 0
0 + 12v + v2 = 0 + -24
12v + v2 = 0 + -24

Combine like terms: 0 + -24 = -24
12v + v2 = -24

The v term is 12v.  Take half its coefficient (6).
Square it (36) and add it to both sides.

Add '36' to each side of the equation.
12v + 36 + v2 = -24 + 36

Reorder the terms:
36 + 12v + v2 = -24 + 36

Combine like terms: -24 + 36 = 12
36 + 12v + v2 = 12

Factor a perfect square on the left side:
(v + 6)(v + 6) = 12

Calculate the square root of the right side: 3.464101615

Break this problem into two subproblems by setting 
(v + 6) equal to 3.464101615 and -3.464101615.

Subproblem 1

v + 6 = 3.464101615 Simplifying v + 6 = 3.464101615 Reorder the terms: 6 + v = 3.464101615 Solving 6 + v = 3.464101615 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-6' to each side of the equation. 6 + -6 + v = 3.464101615 + -6 Combine like terms: 6 + -6 = 0 0 + v = 3.464101615 + -6 v = 3.464101615 + -6 Combine like terms: 3.464101615 + -6 = -2.535898385 v = -2.535898385 Simplifying v = -2.535898385

Subproblem 2

v + 6 = -3.464101615 Simplifying v + 6 = -3.464101615 Reorder the terms: 6 + v = -3.464101615 Solving 6 + v = -3.464101615 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-6' to each side of the equation. 6 + -6 + v = -3.464101615 + -6 Combine like terms: 6 + -6 = 0 0 + v = -3.464101615 + -6 v = -3.464101615 + -6 Combine like terms: -3.464101615 + -6 = -9.464101615 v = -9.464101615 Simplifying v = -9.464101615

Solution

The solution to the problem is based on the solutions from the subproblems. v = {-2.535898385, -9.464101615}

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