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Simplifying (4v3 + -3v2 + 2v) + (3v3 + 6v2 + -9v) = 0 Reorder the terms: (2v + -3v2 + 4v3) + (3v3 + 6v2 + -9v) = 0 Remove parenthesis around (2v + -3v2 + 4v3) 2v + -3v2 + 4v3 + (3v3 + 6v2 + -9v) = 0 Reorder the terms: 2v + -3v2 + 4v3 + (-9v + 6v2 + 3v3) = 0 Remove parenthesis around (-9v + 6v2 + 3v3) 2v + -3v2 + 4v3 + -9v + 6v2 + 3v3 = 0 Reorder the terms: 2v + -9v + -3v2 + 6v2 + 4v3 + 3v3 = 0 Combine like terms: 2v + -9v = -7v -7v + -3v2 + 6v2 + 4v3 + 3v3 = 0 Combine like terms: -3v2 + 6v2 = 3v2 -7v + 3v2 + 4v3 + 3v3 = 0 Combine like terms: 4v3 + 3v3 = 7v3 -7v + 3v2 + 7v3 = 0 Solving -7v + 3v2 + 7v3 = 0 Solving for variable 'v'. Factor out the Greatest Common Factor (GCF), 'v'. v(-7 + 3v + 7v2) = 0Subproblem 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 '(-7 + 3v + 7v2)' equal to zero and attempt to solve: Simplifying -7 + 3v + 7v2 = 0 Solving -7 + 3v + 7v2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -1 + 0.4285714286v + v2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + 0.4285714286v + 1 + v2 = 0 + 1 Reorder the terms: -1 + 1 + 0.4285714286v + v2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + 0.4285714286v + v2 = 0 + 1 0.4285714286v + v2 = 0 + 1 Combine like terms: 0 + 1 = 1 0.4285714286v + v2 = 1 The v term is 0.4285714286v. Take half its coefficient (0.2142857143). Square it (0.04591836735) and add it to both sides. Add '0.04591836735' to each side of the equation. 0.4285714286v + 0.04591836735 + v2 = 1 + 0.04591836735 Reorder the terms: 0.04591836735 + 0.4285714286v + v2 = 1 + 0.04591836735 Combine like terms: 1 + 0.04591836735 = 1.04591836735 0.04591836735 + 0.4285714286v + v2 = 1.04591836735 Factor a perfect square on the left side: (v + 0.2142857143)(v + 0.2142857143) = 1.04591836735 Calculate the square root of the right side: 1.022701505 Break this problem into two subproblems by setting (v + 0.2142857143) equal to 1.022701505 and -1.022701505.Subproblem 1
v + 0.2142857143 = 1.022701505 Simplifying v + 0.2142857143 = 1.022701505 Reorder the terms: 0.2142857143 + v = 1.022701505 Solving 0.2142857143 + v = 1.022701505 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-0.2142857143' to each side of the equation. 0.2142857143 + -0.2142857143 + v = 1.022701505 + -0.2142857143 Combine like terms: 0.2142857143 + -0.2142857143 = 0.0000000000 0.0000000000 + v = 1.022701505 + -0.2142857143 v = 1.022701505 + -0.2142857143 Combine like terms: 1.022701505 + -0.2142857143 = 0.8084157907 v = 0.8084157907 Simplifying v = 0.8084157907Subproblem 2
v + 0.2142857143 = -1.022701505 Simplifying v + 0.2142857143 = -1.022701505 Reorder the terms: 0.2142857143 + v = -1.022701505 Solving 0.2142857143 + v = -1.022701505 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-0.2142857143' to each side of the equation. 0.2142857143 + -0.2142857143 + v = -1.022701505 + -0.2142857143 Combine like terms: 0.2142857143 + -0.2142857143 = 0.0000000000 0.0000000000 + v = -1.022701505 + -0.2142857143 v = -1.022701505 + -0.2142857143 Combine like terms: -1.022701505 + -0.2142857143 = -1.2369872193 v = -1.2369872193 Simplifying v = -1.2369872193Solution
The solution to the problem is based on the solutions from the subproblems. v = {0.8084157907, -1.2369872193}Solution
v = {0, 0.8084157907, -1.2369872193}
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