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Simplifying 2v2 + 8v + 7 = 0 Reorder the terms: 7 + 8v + 2v2 = 0 Solving 7 + 8v + 2v2 = 0 Solving for variable 'v'. Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 3.5 + 4v + v2 = 0 Move the constant term to the right: Add '-3.5' to each side of the equation. 3.5 + 4v + -3.5 + v2 = 0 + -3.5 Reorder the terms: 3.5 + -3.5 + 4v + v2 = 0 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + 4v + v2 = 0 + -3.5 4v + v2 = 0 + -3.5 Combine like terms: 0 + -3.5 = -3.5 4v + v2 = -3.5 The v term is 4v. Take half its coefficient (2). Square it (4) and add it to both sides. Add '4' to each side of the equation. 4v + 4 + v2 = -3.5 + 4 Reorder the terms: 4 + 4v + v2 = -3.5 + 4 Combine like terms: -3.5 + 4 = 0.5 4 + 4v + v2 = 0.5 Factor a perfect square on the left side: (v + 2)(v + 2) = 0.5 Calculate the square root of the right side: 0.707106781 Break this problem into two subproblems by setting (v + 2) equal to 0.707106781 and -0.707106781.Subproblem 1
v + 2 = 0.707106781 Simplifying v + 2 = 0.707106781 Reorder the terms: 2 + v = 0.707106781 Solving 2 + v = 0.707106781 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + v = 0.707106781 + -2 Combine like terms: 2 + -2 = 0 0 + v = 0.707106781 + -2 v = 0.707106781 + -2 Combine like terms: 0.707106781 + -2 = -1.292893219 v = -1.292893219 Simplifying v = -1.292893219Subproblem 2
v + 2 = -0.707106781 Simplifying v + 2 = -0.707106781 Reorder the terms: 2 + v = -0.707106781 Solving 2 + v = -0.707106781 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + v = -0.707106781 + -2 Combine like terms: 2 + -2 = 0 0 + v = -0.707106781 + -2 v = -0.707106781 + -2 Combine like terms: -0.707106781 + -2 = -2.707106781 v = -2.707106781 Simplifying v = -2.707106781Solution
The solution to the problem is based on the solutions from the subproblems. v = {-1.292893219, -2.707106781}
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