4v^2-9v-1=0

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Solution for 4v^2-9v-1=0 equation: 


Simplifying
4v2 + -9v + -1 = 0

Reorder the terms:
-1 + -9v + 4v2 = 0

Solving
-1 + -9v + 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'.
-0.25 + -2.25v + v2 = 0

Move the constant term to the right:

Add '0.25' to each side of the equation.
-0.25 + -2.25v + 0.25 + v2 = 0 + 0.25

Reorder the terms:
-0.25 + 0.25 + -2.25v + v2 = 0 + 0.25

Combine like terms: -0.25 + 0.25 = 0.00
0.00 + -2.25v + v2 = 0 + 0.25
-2.25v + v2 = 0 + 0.25

Combine like terms: 0 + 0.25 = 0.25
-2.25v + v2 = 0.25

The v term is -2.25v.  Take half its coefficient (-1.125).
Square it (1.265625) and add it to both sides.

Add '1.265625' to each side of the equation.
-2.25v + 1.265625 + v2 = 0.25 + 1.265625

Reorder the terms:
1.265625 + -2.25v + v2 = 0.25 + 1.265625

Combine like terms: 0.25 + 1.265625 = 1.515625
1.265625 + -2.25v + v2 = 1.515625

Factor a perfect square on the left side:
(v + -1.125)(v + -1.125) = 1.515625

Calculate the square root of the right side: 1.231107225

Break this problem into two subproblems by setting 
(v + -1.125) equal to 1.231107225 and -1.231107225.


Subproblem 1
v + -1.125 = 1.231107225

Simplifying
v + -1.125 = 1.231107225

Reorder the terms:
-1.125 + v = 1.231107225

Solving
-1.125 + v = 1.231107225

Solving for variable 'v'.

Move all terms containing v to the left, all other terms to the right.

Add '1.125' to each side of the equation.
-1.125 + 1.125 + v = 1.231107225 + 1.125

Combine like terms: -1.125 + 1.125 = 0.000
0.000 + v = 1.231107225 + 1.125
v = 1.231107225 + 1.125

Combine like terms: 1.231107225 + 1.125 = 2.356107225
v = 2.356107225

Simplifying
v = 2.356107225


Subproblem 2
v + -1.125 = -1.231107225

Simplifying
v + -1.125 = -1.231107225

Reorder the terms:
-1.125 + v = -1.231107225

Solving
-1.125 + v = -1.231107225

Solving for variable 'v'.

Move all terms containing v to the left, all other terms to the right.

Add '1.125' to each side of the equation.
-1.125 + 1.125 + v = -1.231107225 + 1.125

Combine like terms: -1.125 + 1.125 = 0.000
0.000 + v = -1.231107225 + 1.125
v = -1.231107225 + 1.125

Combine like terms: -1.231107225 + 1.125 = -0.106107225
v = -0.106107225

Simplifying
v = -0.106107225


Solution
The solution to the problem is based on the solutions
from the subproblems.
v = {2.356107225, -0.106107225}

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