12v^2+9v-15=0

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Solution for 12v^2+9v-15=0 equation: 


Simplifying
12v2 + 9v + -15 = 0

Reorder the terms:
-15 + 9v + 12v2 = 0

Solving
-15 + 9v + 12v2 = 0

Solving for variable 'v'.

Factor out the Greatest Common Factor (GCF), '3'.
3(-5 + 3v + 4v2) = 0

Ignore the factor 3.

Subproblem 1
Set the factor '(-5 + 3v + 4v2)' equal to zero and attempt to solve:

Simplifying
-5 + 3v + 4v2 = 0

Solving
-5 + 3v + 4v2 = 0

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

Divide each side by '4'.
-1.25 + 0.75v + v2 = 0

Move the constant term to the right:

Add '1.25' to each side of the equation.
-1.25 + 0.75v + 1.25 + v2 = 0 + 1.25

Reorder the terms:
-1.25 + 1.25 + 0.75v + v2 = 0 + 1.25

Combine like terms: -1.25 + 1.25 = 0.00
0.00 + 0.75v + v2 = 0 + 1.25
0.75v + v2 = 0 + 1.25

Combine like terms: 0 + 1.25 = 1.25
0.75v + v2 = 1.25

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 = 1.25 + 0.140625

Reorder the terms:
0.140625 + 0.75v + v2 = 1.25 + 0.140625

Combine like terms: 1.25 + 0.140625 = 1.390625
0.140625 + 0.75v + v2 = 1.390625

Factor a perfect square on the left side:
(v + 0.375)(v + 0.375) = 1.390625

Calculate the square root of the right side: 1.179247642

Break this problem into two subproblems by setting 
(v + 0.375) equal to 1.179247642 and -1.179247642.


Subproblem 1
v + 0.375 = 1.179247642

Simplifying
v + 0.375 = 1.179247642

Reorder the terms:
0.375 + v = 1.179247642

Solving
0.375 + v = 1.179247642

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 = 1.179247642 + -0.375

Combine like terms: 0.375 + -0.375 = 0.000
0.000 + v = 1.179247642 + -0.375
v = 1.179247642 + -0.375

Combine like terms: 1.179247642 + -0.375 = 0.804247642
v = 0.804247642

Simplifying
v = 0.804247642


Subproblem 2
v + 0.375 = -1.179247642

Simplifying
v + 0.375 = -1.179247642

Reorder the terms:
0.375 + v = -1.179247642

Solving
0.375 + v = -1.179247642

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 = -1.179247642 + -0.375

Combine like terms: 0.375 + -0.375 = 0.000
0.000 + v = -1.179247642 + -0.375
v = -1.179247642 + -0.375

Combine like terms: -1.179247642 + -0.375 = -1.554247642
v = -1.554247642

Simplifying
v = -1.554247642


Solution
The solution to the problem is based on the solutions
from the subproblems.
v = {0.804247642, -1.554247642}
Solution
v = {0.804247642, -1.554247642}

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