v(7v-27)=-10

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Solution for v(7v-27)=-10 equation:


Simplifying
v(7v + -27) = -10

Reorder the terms:
v(-27 + 7v) = -10
(-27 * v + 7v * v) = -10
(-27v + 7v2) = -10

Solving
-27v + 7v2 = -10

Solving for variable 'v'.

Reorder the terms:
10 + -27v + 7v2 = -10 + 10

Combine like terms: -10 + 10 = 0
10 + -27v + 7v2 = 0

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

Divide each side by '7'.
1.428571429 + -3.857142857v + v2 = 0

Move the constant term to the right:

Add '-1.428571429' to each side of the equation.
1.428571429 + -3.857142857v + -1.428571429 + v2 = 0 + -1.428571429

Reorder the terms:
1.428571429 + -1.428571429 + -3.857142857v + v2 = 0 + -1.428571429

Combine like terms: 1.428571429 + -1.428571429 = 0.000000000
0.000000000 + -3.857142857v + v2 = 0 + -1.428571429
-3.857142857v + v2 = 0 + -1.428571429

Combine like terms: 0 + -1.428571429 = -1.428571429
-3.857142857v + v2 = -1.428571429

The v term is -3.857142857v.  Take half its coefficient (-1.928571429).
Square it (3.719387757) and add it to both sides.

Add '3.719387757' to each side of the equation.
-3.857142857v + 3.719387757 + v2 = -1.428571429 + 3.719387757

Reorder the terms:
3.719387757 + -3.857142857v + v2 = -1.428571429 + 3.719387757

Combine like terms: -1.428571429 + 3.719387757 = 2.290816328
3.719387757 + -3.857142857v + v2 = 2.290816328

Factor a perfect square on the left side:
(v + -1.928571429)(v + -1.928571429) = 2.290816328

Calculate the square root of the right side: 1.513544293

Break this problem into two subproblems by setting 
(v + -1.928571429) equal to 1.513544293 and -1.513544293.

Subproblem 1

v + -1.928571429 = 1.513544293 Simplifying v + -1.928571429 = 1.513544293 Reorder the terms: -1.928571429 + v = 1.513544293 Solving -1.928571429 + v = 1.513544293 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '1.928571429' to each side of the equation. -1.928571429 + 1.928571429 + v = 1.513544293 + 1.928571429 Combine like terms: -1.928571429 + 1.928571429 = 0.000000000 0.000000000 + v = 1.513544293 + 1.928571429 v = 1.513544293 + 1.928571429 Combine like terms: 1.513544293 + 1.928571429 = 3.442115722 v = 3.442115722 Simplifying v = 3.442115722

Subproblem 2

v + -1.928571429 = -1.513544293 Simplifying v + -1.928571429 = -1.513544293 Reorder the terms: -1.928571429 + v = -1.513544293 Solving -1.928571429 + v = -1.513544293 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '1.928571429' to each side of the equation. -1.928571429 + 1.928571429 + v = -1.513544293 + 1.928571429 Combine like terms: -1.928571429 + 1.928571429 = 0.000000000 0.000000000 + v = -1.513544293 + 1.928571429 v = -1.513544293 + 1.928571429 Combine like terms: -1.513544293 + 1.928571429 = 0.415027136 v = 0.415027136 Simplifying v = 0.415027136

Solution

The solution to the problem is based on the solutions from the subproblems. v = {3.442115722, 0.415027136}

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