v(v-31.8)+(v+23.6)=192.6

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Solution for v(v-31.8)+(v+23.6)=192.6 equation:


Simplifying
v(v + -31.8) + (v + 23.6) = 192.6

Reorder the terms:
v(-31.8 + v) + (v + 23.6) = 192.6
(-31.8 * v + v * v) + (v + 23.6) = 192.6
(-31.8v + v2) + (v + 23.6) = 192.6

Reorder the terms:
-31.8v + v2 + (23.6 + v) = 192.6

Remove parenthesis around (23.6 + v)
-31.8v + v2 + 23.6 + v = 192.6

Reorder the terms:
23.6 + -31.8v + v + v2 = 192.6

Combine like terms: -31.8v + v = -30.8v
23.6 + -30.8v + v2 = 192.6

Solving
23.6 + -30.8v + v2 = 192.6

Solving for variable 'v'.

Reorder the terms:
23.6 + -192.6 + -30.8v + v2 = 192.6 + -192.6

Combine like terms: 23.6 + -192.6 = -169
-169 + -30.8v + v2 = 192.6 + -192.6

Combine like terms: 192.6 + -192.6 = 0.0
-169 + -30.8v + v2 = 0.0

Begin completing the square.

Move the constant term to the right:

Add '169' to each side of the equation.
-169 + -30.8v + 169 + v2 = 0.0 + 169

Reorder the terms:
-169 + 169 + -30.8v + v2 = 0.0 + 169

Combine like terms: -169 + 169 = 0
0 + -30.8v + v2 = 0.0 + 169
-30.8v + v2 = 0.0 + 169

Combine like terms: 0.0 + 169 = 169
-30.8v + v2 = 169

The v term is -30.8v.  Take half its coefficient (-15.4).
Square it (237.16) and add it to both sides.

Add '237.16' to each side of the equation.
-30.8v + 237.16 + v2 = 169 + 237.16

Reorder the terms:
237.16 + -30.8v + v2 = 169 + 237.16

Combine like terms: 169 + 237.16 = 406.16
237.16 + -30.8v + v2 = 406.16

Factor a perfect square on the left side:
(v + -15.4)(v + -15.4) = 406.16

Calculate the square root of the right side: 20.153411622

Break this problem into two subproblems by setting 
(v + -15.4) equal to 20.153411622 and -20.153411622.

Subproblem 1

v + -15.4 = 20.153411622 Simplifying v + -15.4 = 20.153411622 Reorder the terms: -15.4 + v = 20.153411622 Solving -15.4 + v = 20.153411622 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '15.4' to each side of the equation. -15.4 + 15.4 + v = 20.153411622 + 15.4 Combine like terms: -15.4 + 15.4 = 0.0 0.0 + v = 20.153411622 + 15.4 v = 20.153411622 + 15.4 Combine like terms: 20.153411622 + 15.4 = 35.553411622 v = 35.553411622 Simplifying v = 35.553411622

Subproblem 2

v + -15.4 = -20.153411622 Simplifying v + -15.4 = -20.153411622 Reorder the terms: -15.4 + v = -20.153411622 Solving -15.4 + v = -20.153411622 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '15.4' to each side of the equation. -15.4 + 15.4 + v = -20.153411622 + 15.4 Combine like terms: -15.4 + 15.4 = 0.0 0.0 + v = -20.153411622 + 15.4 v = -20.153411622 + 15.4 Combine like terms: -20.153411622 + 15.4 = -4.753411622 v = -4.753411622 Simplifying v = -4.753411622

Solution

The solution to the problem is based on the solutions from the subproblems. v = {35.553411622, -4.753411622}

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