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Simplifying -1(-6 + -2v) + 5v(5v + -8) = 0 (-6 * -1 + -2v * -1) + 5v(5v + -8) = 0 (6 + 2v) + 5v(5v + -8) = 0 Reorder the terms: 6 + 2v + 5v(-8 + 5v) = 0 6 + 2v + (-8 * 5v + 5v * 5v) = 0 6 + 2v + (-40v + 25v2) = 0 Combine like terms: 2v + -40v = -38v 6 + -38v + 25v2 = 0 Solving 6 + -38v + 25v2 = 0 Solving for variable 'v'. Begin completing the square. Divide all terms by 25 the coefficient of the squared term: Divide each side by '25'. 0.24 + -1.52v + v2 = 0 Move the constant term to the right: Add '-0.24' to each side of the equation. 0.24 + -1.52v + -0.24 + v2 = 0 + -0.24 Reorder the terms: 0.24 + -0.24 + -1.52v + v2 = 0 + -0.24 Combine like terms: 0.24 + -0.24 = 0.00 0.00 + -1.52v + v2 = 0 + -0.24 -1.52v + v2 = 0 + -0.24 Combine like terms: 0 + -0.24 = -0.24 -1.52v + v2 = -0.24 The v term is -1.52v. Take half its coefficient (-0.76). Square it (0.5776) and add it to both sides. Add '0.5776' to each side of the equation. -1.52v + 0.5776 + v2 = -0.24 + 0.5776 Reorder the terms: 0.5776 + -1.52v + v2 = -0.24 + 0.5776 Combine like terms: -0.24 + 0.5776 = 0.3376 0.5776 + -1.52v + v2 = 0.3376 Factor a perfect square on the left side: (v + -0.76)(v + -0.76) = 0.3376 Calculate the square root of the right side: 0.581033562 Break this problem into two subproblems by setting (v + -0.76) equal to 0.581033562 and -0.581033562.Subproblem 1
v + -0.76 = 0.581033562 Simplifying v + -0.76 = 0.581033562 Reorder the terms: -0.76 + v = 0.581033562 Solving -0.76 + v = 0.581033562 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '0.76' to each side of the equation. -0.76 + 0.76 + v = 0.581033562 + 0.76 Combine like terms: -0.76 + 0.76 = 0.00 0.00 + v = 0.581033562 + 0.76 v = 0.581033562 + 0.76 Combine like terms: 0.581033562 + 0.76 = 1.341033562 v = 1.341033562 Simplifying v = 1.341033562Subproblem 2
v + -0.76 = -0.581033562 Simplifying v + -0.76 = -0.581033562 Reorder the terms: -0.76 + v = -0.581033562 Solving -0.76 + v = -0.581033562 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '0.76' to each side of the equation. -0.76 + 0.76 + v = -0.581033562 + 0.76 Combine like terms: -0.76 + 0.76 = 0.00 0.00 + v = -0.581033562 + 0.76 v = -0.581033562 + 0.76 Combine like terms: -0.581033562 + 0.76 = 0.178966438 v = 0.178966438 Simplifying v = 0.178966438Solution
The solution to the problem is based on the solutions from the subproblems. v = {1.341033562, 0.178966438}
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