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Simplifying 2v + 18 = 16 + -4v(v + 7) Reorder the terms: 18 + 2v = 16 + -4v(v + 7) Reorder the terms: 18 + 2v = 16 + -4v(7 + v) 18 + 2v = 16 + (7 * -4v + v * -4v) 18 + 2v = 16 + (-28v + -4v2) Solving 18 + 2v = 16 + -28v + -4v2 Solving for variable 'v'. Reorder the terms: 18 + -16 + 2v + 28v + 4v2 = 16 + -28v + -4v2 + -16 + 28v + 4v2 Combine like terms: 18 + -16 = 2 2 + 2v + 28v + 4v2 = 16 + -28v + -4v2 + -16 + 28v + 4v2 Combine like terms: 2v + 28v = 30v 2 + 30v + 4v2 = 16 + -28v + -4v2 + -16 + 28v + 4v2 Reorder the terms: 2 + 30v + 4v2 = 16 + -16 + -28v + 28v + -4v2 + 4v2 Combine like terms: 16 + -16 = 0 2 + 30v + 4v2 = 0 + -28v + 28v + -4v2 + 4v2 2 + 30v + 4v2 = -28v + 28v + -4v2 + 4v2 Combine like terms: -28v + 28v = 0 2 + 30v + 4v2 = 0 + -4v2 + 4v2 2 + 30v + 4v2 = -4v2 + 4v2 Combine like terms: -4v2 + 4v2 = 0 2 + 30v + 4v2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(1 + 15v + 2v2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(1 + 15v + 2v2)' equal to zero and attempt to solve: Simplifying 1 + 15v + 2v2 = 0 Solving 1 + 15v + 2v2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 0.5 + 7.5v + v2 = 0 Move the constant term to the right: Add '-0.5' to each side of the equation. 0.5 + 7.5v + -0.5 + v2 = 0 + -0.5 Reorder the terms: 0.5 + -0.5 + 7.5v + v2 = 0 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + 7.5v + v2 = 0 + -0.5 7.5v + v2 = 0 + -0.5 Combine like terms: 0 + -0.5 = -0.5 7.5v + v2 = -0.5 The v term is 7.5v. Take half its coefficient (3.75). Square it (14.0625) and add it to both sides. Add '14.0625' to each side of the equation. 7.5v + 14.0625 + v2 = -0.5 + 14.0625 Reorder the terms: 14.0625 + 7.5v + v2 = -0.5 + 14.0625 Combine like terms: -0.5 + 14.0625 = 13.5625 14.0625 + 7.5v + v2 = 13.5625 Factor a perfect square on the left side: (v + 3.75)(v + 3.75) = 13.5625 Calculate the square root of the right side: 3.682729966 Break this problem into two subproblems by setting (v + 3.75) equal to 3.682729966 and -3.682729966.Subproblem 1
v + 3.75 = 3.682729966 Simplifying v + 3.75 = 3.682729966 Reorder the terms: 3.75 + v = 3.682729966 Solving 3.75 + v = 3.682729966 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-3.75' to each side of the equation. 3.75 + -3.75 + v = 3.682729966 + -3.75 Combine like terms: 3.75 + -3.75 = 0.00 0.00 + v = 3.682729966 + -3.75 v = 3.682729966 + -3.75 Combine like terms: 3.682729966 + -3.75 = -0.067270034 v = -0.067270034 Simplifying v = -0.067270034Subproblem 2
v + 3.75 = -3.682729966 Simplifying v + 3.75 = -3.682729966 Reorder the terms: 3.75 + v = -3.682729966 Solving 3.75 + v = -3.682729966 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-3.75' to each side of the equation. 3.75 + -3.75 + v = -3.682729966 + -3.75 Combine like terms: 3.75 + -3.75 = 0.00 0.00 + v = -3.682729966 + -3.75 v = -3.682729966 + -3.75 Combine like terms: -3.682729966 + -3.75 = -7.432729966 v = -7.432729966 Simplifying v = -7.432729966Solution
The solution to the problem is based on the solutions from the subproblems. v = {-0.067270034, -7.432729966}Solution
v = {-0.067270034, -7.432729966}
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