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Simplifying 5v(4v2 + 12v + -14) = 0 Reorder the terms: 5v(-14 + 12v + 4v2) = 0 (-14 * 5v + 12v * 5v + 4v2 * 5v) = 0 (-70v + 60v2 + 20v3) = 0 Solving -70v + 60v2 + 20v3 = 0 Solving for variable 'v'. Factor out the Greatest Common Factor (GCF), '10v'. 10v(-7 + 6v + 2v2) = 0 Ignore the factor 10.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 '(-7 + 6v + 2v2)' equal to zero and attempt to solve: Simplifying -7 + 6v + 2v2 = 0 Solving -7 + 6v + 2v2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -3.5 + 3v + v2 = 0 Move the constant term to the right: Add '3.5' to each side of the equation. -3.5 + 3v + 3.5 + v2 = 0 + 3.5 Reorder the terms: -3.5 + 3.5 + 3v + v2 = 0 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + 3v + v2 = 0 + 3.5 3v + v2 = 0 + 3.5 Combine like terms: 0 + 3.5 = 3.5 3v + v2 = 3.5 The v term is 3v. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3v + 2.25 + v2 = 3.5 + 2.25 Reorder the terms: 2.25 + 3v + v2 = 3.5 + 2.25 Combine like terms: 3.5 + 2.25 = 5.75 2.25 + 3v + v2 = 5.75 Factor a perfect square on the left side: (v + 1.5)(v + 1.5) = 5.75 Calculate the square root of the right side: 2.397915762 Break this problem into two subproblems by setting (v + 1.5) equal to 2.397915762 and -2.397915762.Subproblem 1
v + 1.5 = 2.397915762 Simplifying v + 1.5 = 2.397915762 Reorder the terms: 1.5 + v = 2.397915762 Solving 1.5 + v = 2.397915762 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + v = 2.397915762 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + v = 2.397915762 + -1.5 v = 2.397915762 + -1.5 Combine like terms: 2.397915762 + -1.5 = 0.897915762 v = 0.897915762 Simplifying v = 0.897915762Subproblem 2
v + 1.5 = -2.397915762 Simplifying v + 1.5 = -2.397915762 Reorder the terms: 1.5 + v = -2.397915762 Solving 1.5 + v = -2.397915762 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + v = -2.397915762 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + v = -2.397915762 + -1.5 v = -2.397915762 + -1.5 Combine like terms: -2.397915762 + -1.5 = -3.897915762 v = -3.897915762 Simplifying v = -3.897915762Solution
The solution to the problem is based on the solutions from the subproblems. v = {0.897915762, -3.897915762}Solution
v = {0, 0.897915762, -3.897915762}
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