0=-4v^2+19v+40

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Solution for 0=-4v^2+19v+40 equation:


Simplifying
0 = -4v2 + 19v + 40

Reorder the terms:
0 = 40 + 19v + -4v2

Solving
0 = 40 + 19v + -4v2

Solving for variable 'v'.

Combine like terms: 0 + -40 = -40
-40 + -19v + 4v2 = 40 + 19v + -4v2 + -40 + -19v + 4v2

Reorder the terms:
-40 + -19v + 4v2 = 40 + -40 + 19v + -19v + -4v2 + 4v2

Combine like terms: 40 + -40 = 0
-40 + -19v + 4v2 = 0 + 19v + -19v + -4v2 + 4v2
-40 + -19v + 4v2 = 19v + -19v + -4v2 + 4v2

Combine like terms: 19v + -19v = 0
-40 + -19v + 4v2 = 0 + -4v2 + 4v2
-40 + -19v + 4v2 = -4v2 + 4v2

Combine like terms: -4v2 + 4v2 = 0
-40 + -19v + 4v2 = 0

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

Divide each side by '4'.
-10 + -4.75v + v2 = 0

Move the constant term to the right:

Add '10' to each side of the equation.
-10 + -4.75v + 10 + v2 = 0 + 10

Reorder the terms:
-10 + 10 + -4.75v + v2 = 0 + 10

Combine like terms: -10 + 10 = 0
0 + -4.75v + v2 = 0 + 10
-4.75v + v2 = 0 + 10

Combine like terms: 0 + 10 = 10
-4.75v + v2 = 10

The v term is -4.75v.  Take half its coefficient (-2.375).
Square it (5.640625) and add it to both sides.

Add '5.640625' to each side of the equation.
-4.75v + 5.640625 + v2 = 10 + 5.640625

Reorder the terms:
5.640625 + -4.75v + v2 = 10 + 5.640625

Combine like terms: 10 + 5.640625 = 15.640625
5.640625 + -4.75v + v2 = 15.640625

Factor a perfect square on the left side:
(v + -2.375)(v + -2.375) = 15.640625

Calculate the square root of the right side: 3.954823005

Break this problem into two subproblems by setting 
(v + -2.375) equal to 3.954823005 and -3.954823005.

Subproblem 1

v + -2.375 = 3.954823005 Simplifying v + -2.375 = 3.954823005 Reorder the terms: -2.375 + v = 3.954823005 Solving -2.375 + v = 3.954823005 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '2.375' to each side of the equation. -2.375 + 2.375 + v = 3.954823005 + 2.375 Combine like terms: -2.375 + 2.375 = 0.000 0.000 + v = 3.954823005 + 2.375 v = 3.954823005 + 2.375 Combine like terms: 3.954823005 + 2.375 = 6.329823005 v = 6.329823005 Simplifying v = 6.329823005

Subproblem 2

v + -2.375 = -3.954823005 Simplifying v + -2.375 = -3.954823005 Reorder the terms: -2.375 + v = -3.954823005 Solving -2.375 + v = -3.954823005 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '2.375' to each side of the equation. -2.375 + 2.375 + v = -3.954823005 + 2.375 Combine like terms: -2.375 + 2.375 = 0.000 0.000 + v = -3.954823005 + 2.375 v = -3.954823005 + 2.375 Combine like terms: -3.954823005 + 2.375 = -1.579823005 v = -1.579823005 Simplifying v = -1.579823005

Solution

The solution to the problem is based on the solutions from the subproblems. v = {6.329823005, -1.579823005}

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