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Simplifying 6v2 = -4 + 12v Solving 6v2 = -4 + 12v Solving for variable 'v'. Reorder the terms: 4 + -12v + 6v2 = -4 + 12v + 4 + -12v Reorder the terms: 4 + -12v + 6v2 = -4 + 4 + 12v + -12v Combine like terms: -4 + 4 = 0 4 + -12v + 6v2 = 0 + 12v + -12v 4 + -12v + 6v2 = 12v + -12v Combine like terms: 12v + -12v = 0 4 + -12v + 6v2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(2 + -6v + 3v2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(2 + -6v + 3v2)' equal to zero and attempt to solve: Simplifying 2 + -6v + 3v2 = 0 Solving 2 + -6v + 3v2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 0.6666666667 + -2v + v2 = 0 Move the constant term to the right: Add '-0.6666666667' to each side of the equation. 0.6666666667 + -2v + -0.6666666667 + v2 = 0 + -0.6666666667 Reorder the terms: 0.6666666667 + -0.6666666667 + -2v + v2 = 0 + -0.6666666667 Combine like terms: 0.6666666667 + -0.6666666667 = 0.0000000000 0.0000000000 + -2v + v2 = 0 + -0.6666666667 -2v + v2 = 0 + -0.6666666667 Combine like terms: 0 + -0.6666666667 = -0.6666666667 -2v + v2 = -0.6666666667 The v term is -2v. Take half its coefficient (-1). Square it (1) and add it to both sides. Add '1' to each side of the equation. -2v + 1 + v2 = -0.6666666667 + 1 Reorder the terms: 1 + -2v + v2 = -0.6666666667 + 1 Combine like terms: -0.6666666667 + 1 = 0.3333333333 1 + -2v + v2 = 0.3333333333 Factor a perfect square on the left side: (v + -1)(v + -1) = 0.3333333333 Calculate the square root of the right side: 0.577350269 Break this problem into two subproblems by setting (v + -1) equal to 0.577350269 and -0.577350269.Subproblem 1
v + -1 = 0.577350269 Simplifying v + -1 = 0.577350269 Reorder the terms: -1 + v = 0.577350269 Solving -1 + v = 0.577350269 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + v = 0.577350269 + 1 Combine like terms: -1 + 1 = 0 0 + v = 0.577350269 + 1 v = 0.577350269 + 1 Combine like terms: 0.577350269 + 1 = 1.577350269 v = 1.577350269 Simplifying v = 1.577350269Subproblem 2
v + -1 = -0.577350269 Simplifying v + -1 = -0.577350269 Reorder the terms: -1 + v = -0.577350269 Solving -1 + v = -0.577350269 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + v = -0.577350269 + 1 Combine like terms: -1 + 1 = 0 0 + v = -0.577350269 + 1 v = -0.577350269 + 1 Combine like terms: -0.577350269 + 1 = 0.422649731 v = 0.422649731 Simplifying v = 0.422649731Solution
The solution to the problem is based on the solutions from the subproblems. v = {1.577350269, 0.422649731}Solution
v = {1.577350269, 0.422649731}
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