4v^2+12v+7=0

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Solution for 4v^2+12v+7=0 equation:


Simplifying
4v2 + 12v + 7 = 0

Reorder the terms:
7 + 12v + 4v2 = 0

Solving
7 + 12v + 4v2 = 0

Solving for variable 'v'.

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

Divide each side by '4'.
1.75 + 3v + v2 = 0

Move the constant term to the right:

Add '-1.75' to each side of the equation.
1.75 + 3v + -1.75 + v2 = 0 + -1.75

Reorder the terms:
1.75 + -1.75 + 3v + v2 = 0 + -1.75

Combine like terms: 1.75 + -1.75 = 0.00
0.00 + 3v + v2 = 0 + -1.75
3v + v2 = 0 + -1.75

Combine like terms: 0 + -1.75 = -1.75
3v + v2 = -1.75

The v term is 3v.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3v + 2.25 + v2 = -1.75 + 2.25

Reorder the terms:
2.25 + 3v + v2 = -1.75 + 2.25

Combine like terms: -1.75 + 2.25 = 0.5
2.25 + 3v + v2 = 0.5

Factor a perfect square on the left side:
(v + 1.5)(v + 1.5) = 0.5

Calculate the square root of the right side: 0.707106781

Break this problem into two subproblems by setting 
(v + 1.5) equal to 0.707106781 and -0.707106781.

Subproblem 1

v + 1.5 = 0.707106781 Simplifying v + 1.5 = 0.707106781 Reorder the terms: 1.5 + v = 0.707106781 Solving 1.5 + v = 0.707106781 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + v = 0.707106781 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + v = 0.707106781 + -1.5 v = 0.707106781 + -1.5 Combine like terms: 0.707106781 + -1.5 = -0.792893219 v = -0.792893219 Simplifying v = -0.792893219

Subproblem 2

v + 1.5 = -0.707106781 Simplifying v + 1.5 = -0.707106781 Reorder the terms: 1.5 + v = -0.707106781 Solving 1.5 + v = -0.707106781 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + v = -0.707106781 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + v = -0.707106781 + -1.5 v = -0.707106781 + -1.5 Combine like terms: -0.707106781 + -1.5 = -2.207106781 v = -2.207106781 Simplifying v = -2.207106781

Solution

The solution to the problem is based on the solutions from the subproblems. v = {-0.792893219, -2.207106781}

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