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Simplifying v(2v + -23) = -19 Reorder the terms: v(-23 + 2v) = -19 (-23 * v + 2v * v) = -19 (-23v + 2v2) = -19 Solving -23v + 2v2 = -19 Solving for variable 'v'. Reorder the terms: 19 + -23v + 2v2 = -19 + 19 Combine like terms: -19 + 19 = 0 19 + -23v + 2v2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 9.5 + -11.5v + v2 = 0 Move the constant term to the right: Add '-9.5' to each side of the equation. 9.5 + -11.5v + -9.5 + v2 = 0 + -9.5 Reorder the terms: 9.5 + -9.5 + -11.5v + v2 = 0 + -9.5 Combine like terms: 9.5 + -9.5 = 0.0 0.0 + -11.5v + v2 = 0 + -9.5 -11.5v + v2 = 0 + -9.5 Combine like terms: 0 + -9.5 = -9.5 -11.5v + v2 = -9.5 The v term is -11.5v. Take half its coefficient (-5.75). Square it (33.0625) and add it to both sides. Add '33.0625' to each side of the equation. -11.5v + 33.0625 + v2 = -9.5 + 33.0625 Reorder the terms: 33.0625 + -11.5v + v2 = -9.5 + 33.0625 Combine like terms: -9.5 + 33.0625 = 23.5625 33.0625 + -11.5v + v2 = 23.5625 Factor a perfect square on the left side: (v + -5.75)(v + -5.75) = 23.5625 Calculate the square root of the right side: 4.85412196 Break this problem into two subproblems by setting (v + -5.75) equal to 4.85412196 and -4.85412196.Subproblem 1
v + -5.75 = 4.85412196 Simplifying v + -5.75 = 4.85412196 Reorder the terms: -5.75 + v = 4.85412196 Solving -5.75 + v = 4.85412196 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '5.75' to each side of the equation. -5.75 + 5.75 + v = 4.85412196 + 5.75 Combine like terms: -5.75 + 5.75 = 0.00 0.00 + v = 4.85412196 + 5.75 v = 4.85412196 + 5.75 Combine like terms: 4.85412196 + 5.75 = 10.60412196 v = 10.60412196 Simplifying v = 10.60412196Subproblem 2
v + -5.75 = -4.85412196 Simplifying v + -5.75 = -4.85412196 Reorder the terms: -5.75 + v = -4.85412196 Solving -5.75 + v = -4.85412196 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '5.75' to each side of the equation. -5.75 + 5.75 + v = -4.85412196 + 5.75 Combine like terms: -5.75 + 5.75 = 0.00 0.00 + v = -4.85412196 + 5.75 v = -4.85412196 + 5.75 Combine like terms: -4.85412196 + 5.75 = 0.89587804 v = 0.89587804 Simplifying v = 0.89587804Solution
The solution to the problem is based on the solutions from the subproblems. v = {10.60412196, 0.89587804}
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