(v+1)=2v^2-2v-20

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Solution for (v+1)=2v^2-2v-20 equation:


Simplifying
(v + 1) = 2v2 + -2v + -20

Reorder the terms:
(1 + v) = 2v2 + -2v + -20

Remove parenthesis around (1 + v)
1 + v = 2v2 + -2v + -20

Reorder the terms:
1 + v = -20 + -2v + 2v2

Solving
1 + v = -20 + -2v + 2v2

Solving for variable 'v'.

Reorder the terms:
1 + 20 + v + 2v + -2v2 = -20 + -2v + 2v2 + 20 + 2v + -2v2

Combine like terms: 1 + 20 = 21
21 + v + 2v + -2v2 = -20 + -2v + 2v2 + 20 + 2v + -2v2

Combine like terms: v + 2v = 3v
21 + 3v + -2v2 = -20 + -2v + 2v2 + 20 + 2v + -2v2

Reorder the terms:
21 + 3v + -2v2 = -20 + 20 + -2v + 2v + 2v2 + -2v2

Combine like terms: -20 + 20 = 0
21 + 3v + -2v2 = 0 + -2v + 2v + 2v2 + -2v2
21 + 3v + -2v2 = -2v + 2v + 2v2 + -2v2

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

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

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

Divide each side by '-2'.
-10.5 + -1.5v + v2 = 0

Move the constant term to the right:

Add '10.5' to each side of the equation.
-10.5 + -1.5v + 10.5 + v2 = 0 + 10.5

Reorder the terms:
-10.5 + 10.5 + -1.5v + v2 = 0 + 10.5

Combine like terms: -10.5 + 10.5 = 0.0
0.0 + -1.5v + v2 = 0 + 10.5
-1.5v + v2 = 0 + 10.5

Combine like terms: 0 + 10.5 = 10.5
-1.5v + v2 = 10.5

The v term is -1.5v.  Take half its coefficient (-0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
-1.5v + 0.5625 + v2 = 10.5 + 0.5625

Reorder the terms:
0.5625 + -1.5v + v2 = 10.5 + 0.5625

Combine like terms: 10.5 + 0.5625 = 11.0625
0.5625 + -1.5v + v2 = 11.0625

Factor a perfect square on the left side:
(v + -0.75)(v + -0.75) = 11.0625

Calculate the square root of the right side: 3.326033674

Break this problem into two subproblems by setting 
(v + -0.75) equal to 3.326033674 and -3.326033674.

Subproblem 1

v + -0.75 = 3.326033674 Simplifying v + -0.75 = 3.326033674 Reorder the terms: -0.75 + v = 3.326033674 Solving -0.75 + v = 3.326033674 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + v = 3.326033674 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + v = 3.326033674 + 0.75 v = 3.326033674 + 0.75 Combine like terms: 3.326033674 + 0.75 = 4.076033674 v = 4.076033674 Simplifying v = 4.076033674

Subproblem 2

v + -0.75 = -3.326033674 Simplifying v + -0.75 = -3.326033674 Reorder the terms: -0.75 + v = -3.326033674 Solving -0.75 + v = -3.326033674 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + v = -3.326033674 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + v = -3.326033674 + 0.75 v = -3.326033674 + 0.75 Combine like terms: -3.326033674 + 0.75 = -2.576033674 v = -2.576033674 Simplifying v = -2.576033674

Solution

The solution to the problem is based on the solutions from the subproblems. v = {4.076033674, -2.576033674}

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