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Simplifying 6v3 + -14v2 + 4v = 2v Reorder the terms: 4v + -14v2 + 6v3 = 2v Solving 4v + -14v2 + 6v3 = 2v Solving for variable 'v'. Reorder the terms: 4v + -2v + -14v2 + 6v3 = 2v + -2v Combine like terms: 4v + -2v = 2v 2v + -14v2 + 6v3 = 2v + -2v Combine like terms: 2v + -2v = 0 2v + -14v2 + 6v3 = 0 Factor out the Greatest Common Factor (GCF), '2v'. 2v(1 + -7v + 3v2) = 0 Ignore the factor 2.Subproblem 1
Set the factor 'v' equal to zero and attempt to solve: Simplifying v = 0 Solving v = 0 Move all terms containing v to the left, all other terms to the right. Simplifying v = 0Subproblem 2
Set the factor '(1 + -7v + 3v2)' equal to zero and attempt to solve: Simplifying 1 + -7v + 3v2 = 0 Solving 1 + -7v + 3v2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 0.3333333333 + -2.333333333v + v2 = 0 Move the constant term to the right: Add '-0.3333333333' to each side of the equation. 0.3333333333 + -2.333333333v + -0.3333333333 + v2 = 0 + -0.3333333333 Reorder the terms: 0.3333333333 + -0.3333333333 + -2.333333333v + v2 = 0 + -0.3333333333 Combine like terms: 0.3333333333 + -0.3333333333 = 0.0000000000 0.0000000000 + -2.333333333v + v2 = 0 + -0.3333333333 -2.333333333v + v2 = 0 + -0.3333333333 Combine like terms: 0 + -0.3333333333 = -0.3333333333 -2.333333333v + v2 = -0.3333333333 The v term is -2.333333333v. Take half its coefficient (-1.166666667). Square it (1.361111112) and add it to both sides. Add '1.361111112' to each side of the equation. -2.333333333v + 1.361111112 + v2 = -0.3333333333 + 1.361111112 Reorder the terms: 1.361111112 + -2.333333333v + v2 = -0.3333333333 + 1.361111112 Combine like terms: -0.3333333333 + 1.361111112 = 1.0277777787 1.361111112 + -2.333333333v + v2 = 1.0277777787 Factor a perfect square on the left side: (v + -1.166666667)(v + -1.166666667) = 1.0277777787 Calculate the square root of the right side: 1.013793756 Break this problem into two subproblems by setting (v + -1.166666667) equal to 1.013793756 and -1.013793756.Subproblem 1
v + -1.166666667 = 1.013793756 Simplifying v + -1.166666667 = 1.013793756 Reorder the terms: -1.166666667 + v = 1.013793756 Solving -1.166666667 + v = 1.013793756 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '1.166666667' to each side of the equation. -1.166666667 + 1.166666667 + v = 1.013793756 + 1.166666667 Combine like terms: -1.166666667 + 1.166666667 = 0.000000000 0.000000000 + v = 1.013793756 + 1.166666667 v = 1.013793756 + 1.166666667 Combine like terms: 1.013793756 + 1.166666667 = 2.180460423 v = 2.180460423 Simplifying v = 2.180460423Subproblem 2
v + -1.166666667 = -1.013793756 Simplifying v + -1.166666667 = -1.013793756 Reorder the terms: -1.166666667 + v = -1.013793756 Solving -1.166666667 + v = -1.013793756 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '1.166666667' to each side of the equation. -1.166666667 + 1.166666667 + v = -1.013793756 + 1.166666667 Combine like terms: -1.166666667 + 1.166666667 = 0.000000000 0.000000000 + v = -1.013793756 + 1.166666667 v = -1.013793756 + 1.166666667 Combine like terms: -1.013793756 + 1.166666667 = 0.152872911 v = 0.152872911 Simplifying v = 0.152872911Solution
The solution to the problem is based on the solutions from the subproblems. v = {2.180460423, 0.152872911}Solution
v = {0, 2.180460423, 0.152872911}
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