15v^2-31v-224=0

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Solution for 15v^2-31v-224=0 equation: 


Simplifying
15v2 + -31v + -224 = 0

Reorder the terms:
-224 + -31v + 15v2 = 0

Solving
-224 + -31v + 15v2 = 0

Solving for variable 'v'.

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

Divide each side by '15'.
-14.93333333 + -2.066666667v + v2 = 0

Move the constant term to the right:

Add '14.93333333' to each side of the equation.
-14.93333333 + -2.066666667v + 14.93333333 + v2 = 0 + 14.93333333

Reorder the terms:
-14.93333333 + 14.93333333 + -2.066666667v + v2 = 0 + 14.93333333

Combine like terms: -14.93333333 + 14.93333333 = 0.00000000
0.00000000 + -2.066666667v + v2 = 0 + 14.93333333
-2.066666667v + v2 = 0 + 14.93333333

Combine like terms: 0 + 14.93333333 = 14.93333333
-2.066666667v + v2 = 14.93333333

The v term is -2.066666667v.  Take half its coefficient (-1.033333334).
Square it (1.067777779) and add it to both sides.

Add '1.067777779' to each side of the equation.
-2.066666667v + 1.067777779 + v2 = 14.93333333 + 1.067777779

Reorder the terms:
1.067777779 + -2.066666667v + v2 = 14.93333333 + 1.067777779

Combine like terms: 14.93333333 + 1.067777779 = 16.001111109
1.067777779 + -2.066666667v + v2 = 16.001111109

Factor a perfect square on the left side:
(v + -1.033333334)(v + -1.033333334) = 16.001111109

Calculate the square root of the right side: 4.000138886

Break this problem into two subproblems by setting 
(v + -1.033333334) equal to 4.000138886 and -4.000138886.


Subproblem 1
v + -1.033333334 = 4.000138886

Simplifying
v + -1.033333334 = 4.000138886

Reorder the terms:
-1.033333334 + v = 4.000138886

Solving
-1.033333334 + v = 4.000138886

Solving for variable 'v'.

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

Add '1.033333334' to each side of the equation.
-1.033333334 + 1.033333334 + v = 4.000138886 + 1.033333334

Combine like terms: -1.033333334 + 1.033333334 = 0.000000000
0.000000000 + v = 4.000138886 + 1.033333334
v = 4.000138886 + 1.033333334

Combine like terms: 4.000138886 + 1.033333334 = 5.03347222
v = 5.03347222

Simplifying
v = 5.03347222


Subproblem 2
v + -1.033333334 = -4.000138886

Simplifying
v + -1.033333334 = -4.000138886

Reorder the terms:
-1.033333334 + v = -4.000138886

Solving
-1.033333334 + v = -4.000138886

Solving for variable 'v'.

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

Add '1.033333334' to each side of the equation.
-1.033333334 + 1.033333334 + v = -4.000138886 + 1.033333334

Combine like terms: -1.033333334 + 1.033333334 = 0.000000000
0.000000000 + v = -4.000138886 + 1.033333334
v = -4.000138886 + 1.033333334

Combine like terms: -4.000138886 + 1.033333334 = -2.966805552
v = -2.966805552

Simplifying
v = -2.966805552


Solution
The solution to the problem is based on the solutions
from the subproblems.
v = {5.03347222, -2.966805552}

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