v^2-v=6v-11

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Solution for v^2-v=6v-11 equation:


Simplifying
v2 + -1v = 6v + -11

Reorder the terms:
-1v + v2 = 6v + -11

Reorder the terms:
-1v + v2 = -11 + 6v

Solving
-1v + v2 = -11 + 6v

Solving for variable 'v'.

Reorder the terms:
11 + -1v + -6v + v2 = -11 + 6v + 11 + -6v

Combine like terms: -1v + -6v = -7v
11 + -7v + v2 = -11 + 6v + 11 + -6v

Reorder the terms:
11 + -7v + v2 = -11 + 11 + 6v + -6v

Combine like terms: -11 + 11 = 0
11 + -7v + v2 = 0 + 6v + -6v
11 + -7v + v2 = 6v + -6v

Combine like terms: 6v + -6v = 0
11 + -7v + v2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-11' to each side of the equation.
11 + -7v + -11 + v2 = 0 + -11

Reorder the terms:
11 + -11 + -7v + v2 = 0 + -11

Combine like terms: 11 + -11 = 0
0 + -7v + v2 = 0 + -11
-7v + v2 = 0 + -11

Combine like terms: 0 + -11 = -11
-7v + v2 = -11

The v term is -7v.  Take half its coefficient (-3.5).
Square it (12.25) and add it to both sides.

Add '12.25' to each side of the equation.
-7v + 12.25 + v2 = -11 + 12.25

Reorder the terms:
12.25 + -7v + v2 = -11 + 12.25

Combine like terms: -11 + 12.25 = 1.25
12.25 + -7v + v2 = 1.25

Factor a perfect square on the left side:
(v + -3.5)(v + -3.5) = 1.25

Calculate the square root of the right side: 1.118033989

Break this problem into two subproblems by setting 
(v + -3.5) equal to 1.118033989 and -1.118033989.

Subproblem 1

v + -3.5 = 1.118033989 Simplifying v + -3.5 = 1.118033989 Reorder the terms: -3.5 + v = 1.118033989 Solving -3.5 + v = 1.118033989 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + v = 1.118033989 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + v = 1.118033989 + 3.5 v = 1.118033989 + 3.5 Combine like terms: 1.118033989 + 3.5 = 4.618033989 v = 4.618033989 Simplifying v = 4.618033989

Subproblem 2

v + -3.5 = -1.118033989 Simplifying v + -3.5 = -1.118033989 Reorder the terms: -3.5 + v = -1.118033989 Solving -3.5 + v = -1.118033989 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + v = -1.118033989 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + v = -1.118033989 + 3.5 v = -1.118033989 + 3.5 Combine like terms: -1.118033989 + 3.5 = 2.381966011 v = 2.381966011 Simplifying v = 2.381966011

Solution

The solution to the problem is based on the solutions from the subproblems. v = {4.618033989, 2.381966011}

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