4v^2-3v-19=0

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Solution for 4v^2-3v-19=0 equation:


Simplifying
4v2 + -3v + -19 = 0

Reorder the terms:
-19 + -3v + 4v2 = 0

Solving
-19 + -3v + 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'.
-4.75 + -0.75v + v2 = 0

Move the constant term to the right:

Add '4.75' to each side of the equation.
-4.75 + -0.75v + 4.75 + v2 = 0 + 4.75

Reorder the terms:
-4.75 + 4.75 + -0.75v + v2 = 0 + 4.75

Combine like terms: -4.75 + 4.75 = 0.00
0.00 + -0.75v + v2 = 0 + 4.75
-0.75v + v2 = 0 + 4.75

Combine like terms: 0 + 4.75 = 4.75
-0.75v + v2 = 4.75

The v term is -0.75v.  Take half its coefficient (-0.375).
Square it (0.140625) and add it to both sides.

Add '0.140625' to each side of the equation.
-0.75v + 0.140625 + v2 = 4.75 + 0.140625

Reorder the terms:
0.140625 + -0.75v + v2 = 4.75 + 0.140625

Combine like terms: 4.75 + 0.140625 = 4.890625
0.140625 + -0.75v + v2 = 4.890625

Factor a perfect square on the left side:
(v + -0.375)(v + -0.375) = 4.890625

Calculate the square root of the right side: 2.211475752

Break this problem into two subproblems by setting 
(v + -0.375) equal to 2.211475752 and -2.211475752.

Subproblem 1

v + -0.375 = 2.211475752 Simplifying v + -0.375 = 2.211475752 Reorder the terms: -0.375 + v = 2.211475752 Solving -0.375 + v = 2.211475752 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '0.375' to each side of the equation. -0.375 + 0.375 + v = 2.211475752 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + v = 2.211475752 + 0.375 v = 2.211475752 + 0.375 Combine like terms: 2.211475752 + 0.375 = 2.586475752 v = 2.586475752 Simplifying v = 2.586475752

Subproblem 2

v + -0.375 = -2.211475752 Simplifying v + -0.375 = -2.211475752 Reorder the terms: -0.375 + v = -2.211475752 Solving -0.375 + v = -2.211475752 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '0.375' to each side of the equation. -0.375 + 0.375 + v = -2.211475752 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + v = -2.211475752 + 0.375 v = -2.211475752 + 0.375 Combine like terms: -2.211475752 + 0.375 = -1.836475752 v = -1.836475752 Simplifying v = -1.836475752

Solution

The solution to the problem is based on the solutions from the subproblems. v = {2.586475752, -1.836475752}

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