8v^2=3v+6

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Solution for 8v^2=3v+6 equation:


Simplifying
8v2 = 3v + 6

Reorder the terms:
8v2 = 6 + 3v

Solving
8v2 = 6 + 3v

Solving for variable 'v'.

Reorder the terms:
-6 + -3v + 8v2 = 6 + 3v + -6 + -3v

Reorder the terms:
-6 + -3v + 8v2 = 6 + -6 + 3v + -3v

Combine like terms: 6 + -6 = 0
-6 + -3v + 8v2 = 0 + 3v + -3v
-6 + -3v + 8v2 = 3v + -3v

Combine like terms: 3v + -3v = 0
-6 + -3v + 8v2 = 0

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

Divide each side by '8'.
-0.75 + -0.375v + v2 = 0

Move the constant term to the right:

Add '0.75' to each side of the equation.
-0.75 + -0.375v + 0.75 + v2 = 0 + 0.75

Reorder the terms:
-0.75 + 0.75 + -0.375v + v2 = 0 + 0.75

Combine like terms: -0.75 + 0.75 = 0.00
0.00 + -0.375v + v2 = 0 + 0.75
-0.375v + v2 = 0 + 0.75

Combine like terms: 0 + 0.75 = 0.75
-0.375v + v2 = 0.75

The v term is -0.375v.  Take half its coefficient (-0.1875).
Square it (0.03515625) and add it to both sides.

Add '0.03515625' to each side of the equation.
-0.375v + 0.03515625 + v2 = 0.75 + 0.03515625

Reorder the terms:
0.03515625 + -0.375v + v2 = 0.75 + 0.03515625

Combine like terms: 0.75 + 0.03515625 = 0.78515625
0.03515625 + -0.375v + v2 = 0.78515625

Factor a perfect square on the left side:
(v + -0.1875)(v + -0.1875) = 0.78515625

Calculate the square root of the right side: 0.88609043

Break this problem into two subproblems by setting 
(v + -0.1875) equal to 0.88609043 and -0.88609043.

Subproblem 1

v + -0.1875 = 0.88609043 Simplifying v + -0.1875 = 0.88609043 Reorder the terms: -0.1875 + v = 0.88609043 Solving -0.1875 + v = 0.88609043 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '0.1875' to each side of the equation. -0.1875 + 0.1875 + v = 0.88609043 + 0.1875 Combine like terms: -0.1875 + 0.1875 = 0.0000 0.0000 + v = 0.88609043 + 0.1875 v = 0.88609043 + 0.1875 Combine like terms: 0.88609043 + 0.1875 = 1.07359043 v = 1.07359043 Simplifying v = 1.07359043

Subproblem 2

v + -0.1875 = -0.88609043 Simplifying v + -0.1875 = -0.88609043 Reorder the terms: -0.1875 + v = -0.88609043 Solving -0.1875 + v = -0.88609043 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '0.1875' to each side of the equation. -0.1875 + 0.1875 + v = -0.88609043 + 0.1875 Combine like terms: -0.1875 + 0.1875 = 0.0000 0.0000 + v = -0.88609043 + 0.1875 v = -0.88609043 + 0.1875 Combine like terms: -0.88609043 + 0.1875 = -0.69859043 v = -0.69859043 Simplifying v = -0.69859043

Solution

The solution to the problem is based on the solutions from the subproblems. v = {1.07359043, -0.69859043}

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