9v^2-v-5=0

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Solution for 9v^2-v-5=0 equation:


Simplifying
9v2 + -1v + -5 = 0

Reorder the terms:
-5 + -1v + 9v2 = 0

Solving
-5 + -1v + 9v2 = 0

Solving for variable 'v'.

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

Divide each side by '9'.
-0.5555555556 + -0.1111111111v + v2 = 0

Move the constant term to the right:

Add '0.5555555556' to each side of the equation.
-0.5555555556 + -0.1111111111v + 0.5555555556 + v2 = 0 + 0.5555555556

Reorder the terms:
-0.5555555556 + 0.5555555556 + -0.1111111111v + v2 = 0 + 0.5555555556

Combine like terms: -0.5555555556 + 0.5555555556 = 0.0000000000
0.0000000000 + -0.1111111111v + v2 = 0 + 0.5555555556
-0.1111111111v + v2 = 0 + 0.5555555556

Combine like terms: 0 + 0.5555555556 = 0.5555555556
-0.1111111111v + v2 = 0.5555555556

The v term is -0.1111111111v.  Take half its coefficient (-0.05555555555).
Square it (0.003086419752) and add it to both sides.

Add '0.003086419752' to each side of the equation.
-0.1111111111v + 0.003086419752 + v2 = 0.5555555556 + 0.003086419752

Reorder the terms:
0.003086419752 + -0.1111111111v + v2 = 0.5555555556 + 0.003086419752

Combine like terms: 0.5555555556 + 0.003086419752 = 0.558641975352
0.003086419752 + -0.1111111111v + v2 = 0.558641975352

Factor a perfect square on the left side:
(v + -0.05555555555)(v + -0.05555555555) = 0.558641975352

Calculate the square root of the right side: 0.747423558

Break this problem into two subproblems by setting 
(v + -0.05555555555) equal to 0.747423558 and -0.747423558.

Subproblem 1

v + -0.05555555555 = 0.747423558 Simplifying v + -0.05555555555 = 0.747423558 Reorder the terms: -0.05555555555 + v = 0.747423558 Solving -0.05555555555 + v = 0.747423558 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '0.05555555555' to each side of the equation. -0.05555555555 + 0.05555555555 + v = 0.747423558 + 0.05555555555 Combine like terms: -0.05555555555 + 0.05555555555 = 0.00000000000 0.00000000000 + v = 0.747423558 + 0.05555555555 v = 0.747423558 + 0.05555555555 Combine like terms: 0.747423558 + 0.05555555555 = 0.80297911355 v = 0.80297911355 Simplifying v = 0.80297911355

Subproblem 2

v + -0.05555555555 = -0.747423558 Simplifying v + -0.05555555555 = -0.747423558 Reorder the terms: -0.05555555555 + v = -0.747423558 Solving -0.05555555555 + v = -0.747423558 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '0.05555555555' to each side of the equation. -0.05555555555 + 0.05555555555 + v = -0.747423558 + 0.05555555555 Combine like terms: -0.05555555555 + 0.05555555555 = 0.00000000000 0.00000000000 + v = -0.747423558 + 0.05555555555 v = -0.747423558 + 0.05555555555 Combine like terms: -0.747423558 + 0.05555555555 = -0.69186800245 v = -0.69186800245 Simplifying v = -0.69186800245

Solution

The solution to the problem is based on the solutions from the subproblems. v = {0.80297911355, -0.69186800245}

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