18v^2+120v+75=0

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Solution for 18v^2+120v+75=0 equation: 


Simplifying
18v2 + 120v + 75 = 0

Reorder the terms:
75 + 120v + 18v2 = 0

Solving
75 + 120v + 18v2 = 0

Solving for variable 'v'.

Factor out the Greatest Common Factor (GCF), '3'.
3(25 + 40v + 6v2) = 0

Ignore the factor 3.

Subproblem 1
Set the factor '(25 + 40v + 6v2)' equal to zero and attempt to solve:

Simplifying
25 + 40v + 6v2 = 0

Solving
25 + 40v + 6v2 = 0

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

Divide each side by '6'.
4.166666667 + 6.666666667v + v2 = 0

Move the constant term to the right:

Add '-4.166666667' to each side of the equation.
4.166666667 + 6.666666667v + -4.166666667 + v2 = 0 + -4.166666667

Reorder the terms:
4.166666667 + -4.166666667 + 6.666666667v + v2 = 0 + -4.166666667

Combine like terms: 4.166666667 + -4.166666667 = 0.000000000
0.000000000 + 6.666666667v + v2 = 0 + -4.166666667
6.666666667v + v2 = 0 + -4.166666667

Combine like terms: 0 + -4.166666667 = -4.166666667
6.666666667v + v2 = -4.166666667

The v term is 6.666666667v.  Take half its coefficient (3.333333334).
Square it (11.11111112) and add it to both sides.

Add '11.11111112' to each side of the equation.
6.666666667v + 11.11111112 + v2 = -4.166666667 + 11.11111112

Reorder the terms:
11.11111112 + 6.666666667v + v2 = -4.166666667 + 11.11111112

Combine like terms: -4.166666667 + 11.11111112 = 6.944444453
11.11111112 + 6.666666667v + v2 = 6.944444453

Factor a perfect square on the left side:
(v + 3.333333334)(v + 3.333333334) = 6.944444453

Calculate the square root of the right side: 2.635231385

Break this problem into two subproblems by setting 
(v + 3.333333334) equal to 2.635231385 and -2.635231385.


Subproblem 1
v + 3.333333334 = 2.635231385

Simplifying
v + 3.333333334 = 2.635231385

Reorder the terms:
3.333333334 + v = 2.635231385

Solving
3.333333334 + v = 2.635231385

Solving for variable 'v'.

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

Add '-3.333333334' to each side of the equation.
3.333333334 + -3.333333334 + v = 2.635231385 + -3.333333334

Combine like terms: 3.333333334 + -3.333333334 = 0.000000000
0.000000000 + v = 2.635231385 + -3.333333334
v = 2.635231385 + -3.333333334

Combine like terms: 2.635231385 + -3.333333334 = -0.698101949
v = -0.698101949

Simplifying
v = -0.698101949


Subproblem 2
v + 3.333333334 = -2.635231385

Simplifying
v + 3.333333334 = -2.635231385

Reorder the terms:
3.333333334 + v = -2.635231385

Solving
3.333333334 + v = -2.635231385

Solving for variable 'v'.

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

Add '-3.333333334' to each side of the equation.
3.333333334 + -3.333333334 + v = -2.635231385 + -3.333333334

Combine like terms: 3.333333334 + -3.333333334 = 0.000000000
0.000000000 + v = -2.635231385 + -3.333333334
v = -2.635231385 + -3.333333334

Combine like terms: -2.635231385 + -3.333333334 = -5.968564719
v = -5.968564719

Simplifying
v = -5.968564719


Solution
The solution to the problem is based on the solutions
from the subproblems.
v = {-0.698101949, -5.968564719}
Solution
v = {-0.698101949, -5.968564719}

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