v(v-30.7)+(v+27.3)=196.7

Solution for v(v-30.7)+(v+27.3)=196.7 equation:

Simplifying
v(v + -30.7) + (v + 27.3) = 196.7

Reorder the terms:
v(-30.7 + v) + (v + 27.3) = 196.7
(-30.7 * v + v * v) + (v + 27.3) = 196.7
(-30.7v + v2) + (v + 27.3) = 196.7

Reorder the terms:
-30.7v + v2 + (27.3 + v) = 196.7

Remove parenthesis around (27.3 + v)
-30.7v + v2 + 27.3 + v = 196.7

Reorder the terms:
27.3 + -30.7v + v + v2 = 196.7

Combine like terms: -30.7v + v = -29.7v
27.3 + -29.7v + v2 = 196.7

Solving
27.3 + -29.7v + v2 = 196.7

Solving for variable 'v'.

Reorder the terms:
27.3 + -196.7 + -29.7v + v2 = 196.7 + -196.7

Combine like terms: 27.3 + -196.7 = -169.4
-169.4 + -29.7v + v2 = 196.7 + -196.7

Combine like terms: 196.7 + -196.7 = 0.0
-169.4 + -29.7v + v2 = 0.0

Begin completing the square.

Move the constant term to the right:

Add '169.4' to each side of the equation.
-169.4 + -29.7v + 169.4 + v2 = 0.0 + 169.4

Reorder the terms:
-169.4 + 169.4 + -29.7v + v2 = 0.0 + 169.4

Combine like terms: -169.4 + 169.4 = 0.0
0.0 + -29.7v + v2 = 0.0 + 169.4
-29.7v + v2 = 0.0 + 169.4

Combine like terms: 0.0 + 169.4 = 169.4
-29.7v + v2 = 169.4

The v term is -29.7v.  Take half its coefficient (-14.85).
Square it (220.5225) and add it to both sides.

Add '220.5225' to each side of the equation.
-29.7v + 220.5225 + v2 = 169.4 + 220.5225

Reorder the terms:
220.5225 + -29.7v + v2 = 169.4 + 220.5225

Combine like terms: 169.4 + 220.5225 = 389.9225
220.5225 + -29.7v + v2 = 389.9225

Factor a perfect square on the left side:
(v + -14.85)(v + -14.85) = 389.9225

Calculate the square root of the right side: 19.746455378

Break this problem into two subproblems by setting 
(v + -14.85) equal to 19.746455378 and -19.746455378.

Subproblem 1
v + -14.85 = 19.746455378

Simplifying
v + -14.85 = 19.746455378

Reorder the terms:
-14.85 + v = 19.746455378

Solving
-14.85 + v = 19.746455378

Solving for variable 'v'.

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

Add '14.85' to each side of the equation.
-14.85 + 14.85 + v = 19.746455378 + 14.85

Combine like terms: -14.85 + 14.85 = 0.00
0.00 + v = 19.746455378 + 14.85
v = 19.746455378 + 14.85

Combine like terms: 19.746455378 + 14.85 = 34.596455378
v = 34.596455378

Simplifying
v = 34.596455378

Subproblem 2
v + -14.85 = -19.746455378

Simplifying
v + -14.85 = -19.746455378

Reorder the terms:
-14.85 + v = -19.746455378

Solving
-14.85 + v = -19.746455378

Solving for variable 'v'.

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

Add '14.85' to each side of the equation.
-14.85 + 14.85 + v = -19.746455378 + 14.85

Combine like terms: -14.85 + 14.85 = 0.00
0.00 + v = -19.746455378 + 14.85
v = -19.746455378 + 14.85

Combine like terms: -19.746455378 + 14.85 = -4.896455378
v = -4.896455378

Simplifying
v = -4.896455378

Solution
The solution to the problem is based on the solutions
from the subproblems.
v = {34.596455378, -4.896455378}