(4v^2-4v+7)-(9v^2+7v-5)+(3v^2+7v+9)=0

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Solution for (4v^2-4v+7)-(9v^2+7v-5)+(3v^2+7v+9)=0 equation:


Simplifying
(4v2 + -4v + 7) + -1(9v2 + 7v + -5) + (3v2 + 7v + 9) = 0

Reorder the terms:
(7 + -4v + 4v2) + -1(9v2 + 7v + -5) + (3v2 + 7v + 9) = 0

Remove parenthesis around (7 + -4v + 4v2)
7 + -4v + 4v2 + -1(9v2 + 7v + -5) + (3v2 + 7v + 9) = 0

Reorder the terms:
7 + -4v + 4v2 + -1(-5 + 7v + 9v2) + (3v2 + 7v + 9) = 0
7 + -4v + 4v2 + (-5 * -1 + 7v * -1 + 9v2 * -1) + (3v2 + 7v + 9) = 0
7 + -4v + 4v2 + (5 + -7v + -9v2) + (3v2 + 7v + 9) = 0

Reorder the terms:
7 + -4v + 4v2 + 5 + -7v + -9v2 + (9 + 7v + 3v2) = 0

Remove parenthesis around (9 + 7v + 3v2)
7 + -4v + 4v2 + 5 + -7v + -9v2 + 9 + 7v + 3v2 = 0

Reorder the terms:
7 + 5 + 9 + -4v + -7v + 7v + 4v2 + -9v2 + 3v2 = 0

Combine like terms: 7 + 5 = 12
12 + 9 + -4v + -7v + 7v + 4v2 + -9v2 + 3v2 = 0

Combine like terms: 12 + 9 = 21
21 + -4v + -7v + 7v + 4v2 + -9v2 + 3v2 = 0

Combine like terms: -4v + -7v = -11v
21 + -11v + 7v + 4v2 + -9v2 + 3v2 = 0

Combine like terms: -11v + 7v = -4v
21 + -4v + 4v2 + -9v2 + 3v2 = 0

Combine like terms: 4v2 + -9v2 = -5v2
21 + -4v + -5v2 + 3v2 = 0

Combine like terms: -5v2 + 3v2 = -2v2
21 + -4v + -2v2 = 0

Solving
21 + -4v + -2v2 = 0

Solving for variable 'v'.

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

Divide each side by '-2'.
-10.5 + 2v + v2 = 0

Move the constant term to the right:

Add '10.5' to each side of the equation.
-10.5 + 2v + 10.5 + v2 = 0 + 10.5

Reorder the terms:
-10.5 + 10.5 + 2v + v2 = 0 + 10.5

Combine like terms: -10.5 + 10.5 = 0.0
0.0 + 2v + v2 = 0 + 10.5
2v + v2 = 0 + 10.5

Combine like terms: 0 + 10.5 = 10.5
2v + v2 = 10.5

The v term is 2v.  Take half its coefficient (1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
2v + 1 + v2 = 10.5 + 1

Reorder the terms:
1 + 2v + v2 = 10.5 + 1

Combine like terms: 10.5 + 1 = 11.5
1 + 2v + v2 = 11.5

Factor a perfect square on the left side:
(v + 1)(v + 1) = 11.5

Calculate the square root of the right side: 3.391164992

Break this problem into two subproblems by setting 
(v + 1) equal to 3.391164992 and -3.391164992.

Subproblem 1

v + 1 = 3.391164992 Simplifying v + 1 = 3.391164992 Reorder the terms: 1 + v = 3.391164992 Solving 1 + v = 3.391164992 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + v = 3.391164992 + -1 Combine like terms: 1 + -1 = 0 0 + v = 3.391164992 + -1 v = 3.391164992 + -1 Combine like terms: 3.391164992 + -1 = 2.391164992 v = 2.391164992 Simplifying v = 2.391164992

Subproblem 2

v + 1 = -3.391164992 Simplifying v + 1 = -3.391164992 Reorder the terms: 1 + v = -3.391164992 Solving 1 + v = -3.391164992 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + v = -3.391164992 + -1 Combine like terms: 1 + -1 = 0 0 + v = -3.391164992 + -1 v = -3.391164992 + -1 Combine like terms: -3.391164992 + -1 = -4.391164992 v = -4.391164992 Simplifying v = -4.391164992

Solution

The solution to the problem is based on the solutions from the subproblems. v = {2.391164992, -4.391164992}

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