4v^2+40v+25=v^2+5v

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Solution for 4v^2+40v+25=v^2+5v equation:


Simplifying
4v2 + 40v + 25 = v2 + 5v

Reorder the terms:
25 + 40v + 4v2 = v2 + 5v

Reorder the terms:
25 + 40v + 4v2 = 5v + v2

Solving
25 + 40v + 4v2 = 5v + v2

Solving for variable 'v'.

Reorder the terms:
25 + 40v + -5v + 4v2 + -1v2 = 5v + v2 + -5v + -1v2

Combine like terms: 40v + -5v = 35v
25 + 35v + 4v2 + -1v2 = 5v + v2 + -5v + -1v2

Combine like terms: 4v2 + -1v2 = 3v2
25 + 35v + 3v2 = 5v + v2 + -5v + -1v2

Reorder the terms:
25 + 35v + 3v2 = 5v + -5v + v2 + -1v2

Combine like terms: 5v + -5v = 0
25 + 35v + 3v2 = 0 + v2 + -1v2
25 + 35v + 3v2 = v2 + -1v2

Combine like terms: v2 + -1v2 = 0
25 + 35v + 3v2 = 0

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

Divide each side by '3'.
8.333333333 + 11.66666667v + v2 = 0

Move the constant term to the right:

Add '-8.333333333' to each side of the equation.
8.333333333 + 11.66666667v + -8.333333333 + v2 = 0 + -8.333333333

Reorder the terms:
8.333333333 + -8.333333333 + 11.66666667v + v2 = 0 + -8.333333333

Combine like terms: 8.333333333 + -8.333333333 = 0.000000000
0.000000000 + 11.66666667v + v2 = 0 + -8.333333333
11.66666667v + v2 = 0 + -8.333333333

Combine like terms: 0 + -8.333333333 = -8.333333333
11.66666667v + v2 = -8.333333333

The v term is 11.66666667v.  Take half its coefficient (5.833333335).
Square it (34.02777780) and add it to both sides.

Add '34.02777780' to each side of the equation.
11.66666667v + 34.02777780 + v2 = -8.333333333 + 34.02777780

Reorder the terms:
34.02777780 + 11.66666667v + v2 = -8.333333333 + 34.02777780

Combine like terms: -8.333333333 + 34.02777780 = 25.694444467
34.02777780 + 11.66666667v + v2 = 25.694444467

Factor a perfect square on the left side:
(v + 5.833333335)(v + 5.833333335) = 25.694444467

Calculate the square root of the right side: 5.068968777

Break this problem into two subproblems by setting 
(v + 5.833333335) equal to 5.068968777 and -5.068968777.

Subproblem 1

v + 5.833333335 = 5.068968777 Simplifying v + 5.833333335 = 5.068968777 Reorder the terms: 5.833333335 + v = 5.068968777 Solving 5.833333335 + v = 5.068968777 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-5.833333335' to each side of the equation. 5.833333335 + -5.833333335 + v = 5.068968777 + -5.833333335 Combine like terms: 5.833333335 + -5.833333335 = 0.000000000 0.000000000 + v = 5.068968777 + -5.833333335 v = 5.068968777 + -5.833333335 Combine like terms: 5.068968777 + -5.833333335 = -0.764364558 v = -0.764364558 Simplifying v = -0.764364558

Subproblem 2

v + 5.833333335 = -5.068968777 Simplifying v + 5.833333335 = -5.068968777 Reorder the terms: 5.833333335 + v = -5.068968777 Solving 5.833333335 + v = -5.068968777 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-5.833333335' to each side of the equation. 5.833333335 + -5.833333335 + v = -5.068968777 + -5.833333335 Combine like terms: 5.833333335 + -5.833333335 = 0.000000000 0.000000000 + v = -5.068968777 + -5.833333335 v = -5.068968777 + -5.833333335 Combine like terms: -5.068968777 + -5.833333335 = -10.902302112 v = -10.902302112 Simplifying v = -10.902302112

Solution

The solution to the problem is based on the solutions from the subproblems. v = {-0.764364558, -10.902302112}

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