8(2+2v)+8v(7-2v)=

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Solution for 8(2+2v)+8v(7-2v)= equation: 


Simplifying
8(2 + 2v) + 8v(7 + -2v) = 0
(2 * 8 + 2v * 8) + 8v(7 + -2v) = 0
(16 + 16v) + 8v(7 + -2v) = 0
16 + 16v + (7 * 8v + -2v * 8v) = 0
16 + 16v + (56v + -16v2) = 0

Combine like terms: 16v + 56v = 72v
16 + 72v + -16v2 = 0

Solving
16 + 72v + -16v2 = 0

Solving for variable 'v'.

Factor out the Greatest Common Factor (GCF), '8'.
8(2 + 9v + -2v2) = 0

Ignore the factor 8.

Subproblem 1
Set the factor '(2 + 9v + -2v2)' equal to zero and attempt to solve:

Simplifying
2 + 9v + -2v2 = 0

Solving
2 + 9v + -2v2 = 0

Begin completing the square.  Divide all terms by
-2 the coefficient of the squared term: 

Divide each side by '-2'.
-1 + -4.5v + v2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + -4.5v + 1 + v2 = 0 + 1

Reorder the terms:
-1 + 1 + -4.5v + v2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + -4.5v + v2 = 0 + 1
-4.5v + v2 = 0 + 1

Combine like terms: 0 + 1 = 1
-4.5v + v2 = 1

The v term is -4.5v.  Take half its coefficient (-2.25).
Square it (5.0625) and add it to both sides.

Add '5.0625' to each side of the equation.
-4.5v + 5.0625 + v2 = 1 + 5.0625

Reorder the terms:
5.0625 + -4.5v + v2 = 1 + 5.0625

Combine like terms: 1 + 5.0625 = 6.0625
5.0625 + -4.5v + v2 = 6.0625

Factor a perfect square on the left side:
(v + -2.25)(v + -2.25) = 6.0625

Calculate the square root of the right side: 2.46221445

Break this problem into two subproblems by setting 
(v + -2.25) equal to 2.46221445 and -2.46221445.


Subproblem 1
v + -2.25 = 2.46221445

Simplifying
v + -2.25 = 2.46221445

Reorder the terms:
-2.25 + v = 2.46221445

Solving
-2.25 + v = 2.46221445

Solving for variable 'v'.

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

Add '2.25' to each side of the equation.
-2.25 + 2.25 + v = 2.46221445 + 2.25

Combine like terms: -2.25 + 2.25 = 0.00
0.00 + v = 2.46221445 + 2.25
v = 2.46221445 + 2.25

Combine like terms: 2.46221445 + 2.25 = 4.71221445
v = 4.71221445

Simplifying
v = 4.71221445


Subproblem 2
v + -2.25 = -2.46221445

Simplifying
v + -2.25 = -2.46221445

Reorder the terms:
-2.25 + v = -2.46221445

Solving
-2.25 + v = -2.46221445

Solving for variable 'v'.

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

Add '2.25' to each side of the equation.
-2.25 + 2.25 + v = -2.46221445 + 2.25

Combine like terms: -2.25 + 2.25 = 0.00
0.00 + v = -2.46221445 + 2.25
v = -2.46221445 + 2.25

Combine like terms: -2.46221445 + 2.25 = -0.21221445
v = -0.21221445

Simplifying
v = -0.21221445


Solution
The solution to the problem is based on the solutions
from the subproblems.
v = {4.71221445, -0.21221445}
Solution
v = {4.71221445, -0.21221445}

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