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Simplifying -4v2 + 41v = 14 Reorder the terms: 41v + -4v2 = 14 Solving 41v + -4v2 = 14 Solving for variable 'v'. Reorder the terms: -14 + 41v + -4v2 = 14 + -14 Combine like terms: 14 + -14 = 0 -14 + 41v + -4v2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. 3.5 + -10.25v + v2 = 0 Move the constant term to the right: Add '-3.5' to each side of the equation. 3.5 + -10.25v + -3.5 + v2 = 0 + -3.5 Reorder the terms: 3.5 + -3.5 + -10.25v + v2 = 0 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + -10.25v + v2 = 0 + -3.5 -10.25v + v2 = 0 + -3.5 Combine like terms: 0 + -3.5 = -3.5 -10.25v + v2 = -3.5 The v term is -10.25v. Take half its coefficient (-5.125). Square it (26.265625) and add it to both sides. Add '26.265625' to each side of the equation. -10.25v + 26.265625 + v2 = -3.5 + 26.265625 Reorder the terms: 26.265625 + -10.25v + v2 = -3.5 + 26.265625 Combine like terms: -3.5 + 26.265625 = 22.765625 26.265625 + -10.25v + v2 = 22.765625 Factor a perfect square on the left side: (v + -5.125)(v + -5.125) = 22.765625 Calculate the square root of the right side: 4.771333671 Break this problem into two subproblems by setting (v + -5.125) equal to 4.771333671 and -4.771333671.Subproblem 1
v + -5.125 = 4.771333671 Simplifying v + -5.125 = 4.771333671 Reorder the terms: -5.125 + v = 4.771333671 Solving -5.125 + v = 4.771333671 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '5.125' to each side of the equation. -5.125 + 5.125 + v = 4.771333671 + 5.125 Combine like terms: -5.125 + 5.125 = 0.000 0.000 + v = 4.771333671 + 5.125 v = 4.771333671 + 5.125 Combine like terms: 4.771333671 + 5.125 = 9.896333671 v = 9.896333671 Simplifying v = 9.896333671Subproblem 2
v + -5.125 = -4.771333671 Simplifying v + -5.125 = -4.771333671 Reorder the terms: -5.125 + v = -4.771333671 Solving -5.125 + v = -4.771333671 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '5.125' to each side of the equation. -5.125 + 5.125 + v = -4.771333671 + 5.125 Combine like terms: -5.125 + 5.125 = 0.000 0.000 + v = -4.771333671 + 5.125 v = -4.771333671 + 5.125 Combine like terms: -4.771333671 + 5.125 = 0.353666329 v = 0.353666329 Simplifying v = 0.353666329Solution
The solution to the problem is based on the solutions from the subproblems. v = {9.896333671, 0.353666329}
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