3v^2-7=5v

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Solution for 3v^2-7=5v equation:


Simplifying
3v2 + -7 = 5v

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

Solving
-7 + 3v2 = 5v

Solving for variable 'v'.

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

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

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

Divide each side by '3'.
-2.333333333 + -1.666666667v + v2 = 0

Move the constant term to the right:

Add '2.333333333' to each side of the equation.
-2.333333333 + -1.666666667v + 2.333333333 + v2 = 0 + 2.333333333

Reorder the terms:
-2.333333333 + 2.333333333 + -1.666666667v + v2 = 0 + 2.333333333

Combine like terms: -2.333333333 + 2.333333333 = 0.000000000
0.000000000 + -1.666666667v + v2 = 0 + 2.333333333
-1.666666667v + v2 = 0 + 2.333333333

Combine like terms: 0 + 2.333333333 = 2.333333333
-1.666666667v + v2 = 2.333333333

The v term is -1.666666667v.  Take half its coefficient (-0.8333333335).
Square it (0.6944444447) and add it to both sides.

Add '0.6944444447' to each side of the equation.
-1.666666667v + 0.6944444447 + v2 = 2.333333333 + 0.6944444447

Reorder the terms:
0.6944444447 + -1.666666667v + v2 = 2.333333333 + 0.6944444447

Combine like terms: 2.333333333 + 0.6944444447 = 3.0277777777
0.6944444447 + -1.666666667v + v2 = 3.0277777777

Factor a perfect square on the left side:
(v + -0.8333333335)(v + -0.8333333335) = 3.0277777777

Calculate the square root of the right side: 1.740051085

Break this problem into two subproblems by setting 
(v + -0.8333333335) equal to 1.740051085 and -1.740051085.

Subproblem 1

v + -0.8333333335 = 1.740051085 Simplifying v + -0.8333333335 = 1.740051085 Reorder the terms: -0.8333333335 + v = 1.740051085 Solving -0.8333333335 + v = 1.740051085 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '0.8333333335' to each side of the equation. -0.8333333335 + 0.8333333335 + v = 1.740051085 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + v = 1.740051085 + 0.8333333335 v = 1.740051085 + 0.8333333335 Combine like terms: 1.740051085 + 0.8333333335 = 2.5733844185 v = 2.5733844185 Simplifying v = 2.5733844185

Subproblem 2

v + -0.8333333335 = -1.740051085 Simplifying v + -0.8333333335 = -1.740051085 Reorder the terms: -0.8333333335 + v = -1.740051085 Solving -0.8333333335 + v = -1.740051085 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '0.8333333335' to each side of the equation. -0.8333333335 + 0.8333333335 + v = -1.740051085 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + v = -1.740051085 + 0.8333333335 v = -1.740051085 + 0.8333333335 Combine like terms: -1.740051085 + 0.8333333335 = -0.9067177515 v = -0.9067177515 Simplifying v = -0.9067177515

Solution

The solution to the problem is based on the solutions from the subproblems. v = {2.5733844185, -0.9067177515}

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