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Simplifying 12v2 = 4 + 12v Solving 12v2 = 4 + 12v Solving for variable 'v'. Reorder the terms: -4 + -12v + 12v2 = 4 + 12v + -4 + -12v Reorder the terms: -4 + -12v + 12v2 = 4 + -4 + 12v + -12v Combine like terms: 4 + -4 = 0 -4 + -12v + 12v2 = 0 + 12v + -12v -4 + -12v + 12v2 = 12v + -12v Combine like terms: 12v + -12v = 0 -4 + -12v + 12v2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(-1 + -3v + 3v2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(-1 + -3v + 3v2)' equal to zero and attempt to solve: Simplifying -1 + -3v + 3v2 = 0 Solving -1 + -3v + 3v2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -0.3333333333 + -1v + v2 = 0 Move the constant term to the right: Add '0.3333333333' to each side of the equation. -0.3333333333 + -1v + 0.3333333333 + v2 = 0 + 0.3333333333 Reorder the terms: -0.3333333333 + 0.3333333333 + -1v + v2 = 0 + 0.3333333333 Combine like terms: -0.3333333333 + 0.3333333333 = 0.0000000000 0.0000000000 + -1v + v2 = 0 + 0.3333333333 -1v + v2 = 0 + 0.3333333333 Combine like terms: 0 + 0.3333333333 = 0.3333333333 -1v + v2 = 0.3333333333 The v term is -1v. Take half its coefficient (-0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -1v + 0.25 + v2 = 0.3333333333 + 0.25 Reorder the terms: 0.25 + -1v + v2 = 0.3333333333 + 0.25 Combine like terms: 0.3333333333 + 0.25 = 0.5833333333 0.25 + -1v + v2 = 0.5833333333 Factor a perfect square on the left side: (v + -0.5)(v + -0.5) = 0.5833333333 Calculate the square root of the right side: 0.763762616 Break this problem into two subproblems by setting (v + -0.5) equal to 0.763762616 and -0.763762616.Subproblem 1
v + -0.5 = 0.763762616 Simplifying v + -0.5 = 0.763762616 Reorder the terms: -0.5 + v = 0.763762616 Solving -0.5 + v = 0.763762616 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + v = 0.763762616 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + v = 0.763762616 + 0.5 v = 0.763762616 + 0.5 Combine like terms: 0.763762616 + 0.5 = 1.263762616 v = 1.263762616 Simplifying v = 1.263762616Subproblem 2
v + -0.5 = -0.763762616 Simplifying v + -0.5 = -0.763762616 Reorder the terms: -0.5 + v = -0.763762616 Solving -0.5 + v = -0.763762616 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + v = -0.763762616 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + v = -0.763762616 + 0.5 v = -0.763762616 + 0.5 Combine like terms: -0.763762616 + 0.5 = -0.263762616 v = -0.263762616 Simplifying v = -0.263762616Solution
The solution to the problem is based on the solutions from the subproblems. v = {1.263762616, -0.263762616}Solution
v = {1.263762616, -0.263762616}
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