v(v-29.7)+(v+24.6)=192

Solution for v(v-29.7)+(v+24.6)=192 equation:

Simplifying
v(v + -29.7) + (v + 24.6) = 192

Reorder the terms:
v(-29.7 + v) + (v + 24.6) = 192
(-29.7 * v + v * v) + (v + 24.6) = 192
(-29.7v + v2) + (v + 24.6) = 192

Reorder the terms:
-29.7v + v2 + (24.6 + v) = 192

Remove parenthesis around (24.6 + v)
-29.7v + v2 + 24.6 + v = 192

Reorder the terms:
24.6 + -29.7v + v + v2 = 192

Combine like terms: -29.7v + v = -28.7v
24.6 + -28.7v + v2 = 192

Solving
24.6 + -28.7v + v2 = 192

Solving for variable 'v'.

Reorder the terms:
24.6 + -192 + -28.7v + v2 = 192 + -192

Combine like terms: 24.6 + -192 = -167.4
-167.4 + -28.7v + v2 = 192 + -192

Combine like terms: 192 + -192 = 0
-167.4 + -28.7v + v2 = 0

Begin completing the square.

Move the constant term to the right:

Add '167.4' to each side of the equation.
-167.4 + -28.7v + 167.4 + v2 = 0 + 167.4

Reorder the terms:
-167.4 + 167.4 + -28.7v + v2 = 0 + 167.4

Combine like terms: -167.4 + 167.4 = 0.0
0.0 + -28.7v + v2 = 0 + 167.4
-28.7v + v2 = 0 + 167.4

Combine like terms: 0 + 167.4 = 167.4
-28.7v + v2 = 167.4

The v term is -28.7v.  Take half its coefficient (-14.35).
Square it (205.9225) and add it to both sides.

Add '205.9225' to each side of the equation.
-28.7v + 205.9225 + v2 = 167.4 + 205.9225

Reorder the terms:
205.9225 + -28.7v + v2 = 167.4 + 205.9225

Combine like terms: 167.4 + 205.9225 = 373.3225
205.9225 + -28.7v + v2 = 373.3225

Factor a perfect square on the left side:
(v + -14.35)(v + -14.35) = 373.3225

Calculate the square root of the right side: 19.32155532

Break this problem into two subproblems by setting 
(v + -14.35) equal to 19.32155532 and -19.32155532.

Subproblem 1
v + -14.35 = 19.32155532

Simplifying
v + -14.35 = 19.32155532

Reorder the terms:
-14.35 + v = 19.32155532

Solving
-14.35 + v = 19.32155532

Solving for variable 'v'.

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

Add '14.35' to each side of the equation.
-14.35 + 14.35 + v = 19.32155532 + 14.35

Combine like terms: -14.35 + 14.35 = 0.00
0.00 + v = 19.32155532 + 14.35
v = 19.32155532 + 14.35

Combine like terms: 19.32155532 + 14.35 = 33.67155532
v = 33.67155532

Simplifying
v = 33.67155532

Subproblem 2
v + -14.35 = -19.32155532

Simplifying
v + -14.35 = -19.32155532

Reorder the terms:
-14.35 + v = -19.32155532

Solving
-14.35 + v = -19.32155532

Solving for variable 'v'.

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

Add '14.35' to each side of the equation.
-14.35 + 14.35 + v = -19.32155532 + 14.35

Combine like terms: -14.35 + 14.35 = 0.00
0.00 + v = -19.32155532 + 14.35
v = -19.32155532 + 14.35

Combine like terms: -19.32155532 + 14.35 = -4.97155532
v = -4.97155532

Simplifying
v = -4.97155532

Solution
The solution to the problem is based on the solutions
from the subproblems.
v = {33.67155532, -4.97155532}