5v^2=-31v-28

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Solution for 5v^2=-31v-28 equation:


Simplifying
5v2 = -31v + -28

Reorder the terms:
5v2 = -28 + -31v

Solving
5v2 = -28 + -31v

Solving for variable 'v'.

Reorder the terms:
28 + 31v + 5v2 = -28 + -31v + 28 + 31v

Reorder the terms:
28 + 31v + 5v2 = -28 + 28 + -31v + 31v

Combine like terms: -28 + 28 = 0
28 + 31v + 5v2 = 0 + -31v + 31v
28 + 31v + 5v2 = -31v + 31v

Combine like terms: -31v + 31v = 0
28 + 31v + 5v2 = 0

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

Divide each side by '5'.
5.6 + 6.2v + v2 = 0

Move the constant term to the right:

Add '-5.6' to each side of the equation.
5.6 + 6.2v + -5.6 + v2 = 0 + -5.6

Reorder the terms:
5.6 + -5.6 + 6.2v + v2 = 0 + -5.6

Combine like terms: 5.6 + -5.6 = 0.0
0.0 + 6.2v + v2 = 0 + -5.6
6.2v + v2 = 0 + -5.6

Combine like terms: 0 + -5.6 = -5.6
6.2v + v2 = -5.6

The v term is 6.2v.  Take half its coefficient (3.1).
Square it (9.61) and add it to both sides.

Add '9.61' to each side of the equation.
6.2v + 9.61 + v2 = -5.6 + 9.61

Reorder the terms:
9.61 + 6.2v + v2 = -5.6 + 9.61

Combine like terms: -5.6 + 9.61 = 4.01
9.61 + 6.2v + v2 = 4.01

Factor a perfect square on the left side:
(v + 3.1)(v + 3.1) = 4.01

Calculate the square root of the right side: 2.002498439

Break this problem into two subproblems by setting 
(v + 3.1) equal to 2.002498439 and -2.002498439.

Subproblem 1

v + 3.1 = 2.002498439 Simplifying v + 3.1 = 2.002498439 Reorder the terms: 3.1 + v = 2.002498439 Solving 3.1 + v = 2.002498439 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-3.1' to each side of the equation. 3.1 + -3.1 + v = 2.002498439 + -3.1 Combine like terms: 3.1 + -3.1 = 0.0 0.0 + v = 2.002498439 + -3.1 v = 2.002498439 + -3.1 Combine like terms: 2.002498439 + -3.1 = -1.097501561 v = -1.097501561 Simplifying v = -1.097501561

Subproblem 2

v + 3.1 = -2.002498439 Simplifying v + 3.1 = -2.002498439 Reorder the terms: 3.1 + v = -2.002498439 Solving 3.1 + v = -2.002498439 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-3.1' to each side of the equation. 3.1 + -3.1 + v = -2.002498439 + -3.1 Combine like terms: 3.1 + -3.1 = 0.0 0.0 + v = -2.002498439 + -3.1 v = -2.002498439 + -3.1 Combine like terms: -2.002498439 + -3.1 = -5.102498439 v = -5.102498439 Simplifying v = -5.102498439

Solution

The solution to the problem is based on the solutions from the subproblems. v = {-1.097501561, -5.102498439}

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