v^2+8v+3=-v^2

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Solution for v^2+8v+3=-v^2 equation:


Simplifying
v2 + 8v + 3 = -1v2

Reorder the terms:
3 + 8v + v2 = -1v2

Solving
3 + 8v + v2 = -1v2

Solving for variable 'v'.

Combine like terms: v2 + v2 = 2v2
3 + 8v + 2v2 = -1v2 + v2

Combine like terms: -1v2 + v2 = 0
3 + 8v + 2v2 = 0

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

Divide each side by '2'.
1.5 + 4v + v2 = 0

Move the constant term to the right:

Add '-1.5' to each side of the equation.
1.5 + 4v + -1.5 + v2 = 0 + -1.5

Reorder the terms:
1.5 + -1.5 + 4v + v2 = 0 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + 4v + v2 = 0 + -1.5
4v + v2 = 0 + -1.5

Combine like terms: 0 + -1.5 = -1.5
4v + v2 = -1.5

The v term is 4v.  Take half its coefficient (2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
4v + 4 + v2 = -1.5 + 4

Reorder the terms:
4 + 4v + v2 = -1.5 + 4

Combine like terms: -1.5 + 4 = 2.5
4 + 4v + v2 = 2.5

Factor a perfect square on the left side:
(v + 2)(v + 2) = 2.5

Calculate the square root of the right side: 1.58113883

Break this problem into two subproblems by setting 
(v + 2) equal to 1.58113883 and -1.58113883.

Subproblem 1

v + 2 = 1.58113883 Simplifying v + 2 = 1.58113883 Reorder the terms: 2 + v = 1.58113883 Solving 2 + v = 1.58113883 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + v = 1.58113883 + -2 Combine like terms: 2 + -2 = 0 0 + v = 1.58113883 + -2 v = 1.58113883 + -2 Combine like terms: 1.58113883 + -2 = -0.41886117 v = -0.41886117 Simplifying v = -0.41886117

Subproblem 2

v + 2 = -1.58113883 Simplifying v + 2 = -1.58113883 Reorder the terms: 2 + v = -1.58113883 Solving 2 + v = -1.58113883 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + v = -1.58113883 + -2 Combine like terms: 2 + -2 = 0 0 + v = -1.58113883 + -2 v = -1.58113883 + -2 Combine like terms: -1.58113883 + -2 = -3.58113883 v = -3.58113883 Simplifying v = -3.58113883

Solution

The solution to the problem is based on the solutions from the subproblems. v = {-0.41886117, -3.58113883}

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