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Simplifying 2(-3 + -1v) + -3v(1 + -6v) = 0 (-3 * 2 + -1v * 2) + -3v(1 + -6v) = 0 (-6 + -2v) + -3v(1 + -6v) = 0 -6 + -2v + (1 * -3v + -6v * -3v) = 0 -6 + -2v + (-3v + 18v2) = 0 Combine like terms: -2v + -3v = -5v -6 + -5v + 18v2 = 0 Solving -6 + -5v + 18v2 = 0 Solving for variable 'v'. Begin completing the square. Divide all terms by 18 the coefficient of the squared term: Divide each side by '18'. -0.3333333333 + -0.2777777778v + v2 = 0 Move the constant term to the right: Add '0.3333333333' to each side of the equation. -0.3333333333 + -0.2777777778v + 0.3333333333 + v2 = 0 + 0.3333333333 Reorder the terms: -0.3333333333 + 0.3333333333 + -0.2777777778v + v2 = 0 + 0.3333333333 Combine like terms: -0.3333333333 + 0.3333333333 = 0.0000000000 0.0000000000 + -0.2777777778v + v2 = 0 + 0.3333333333 -0.2777777778v + v2 = 0 + 0.3333333333 Combine like terms: 0 + 0.3333333333 = 0.3333333333 -0.2777777778v + v2 = 0.3333333333 The v term is -0.2777777778v. Take half its coefficient (-0.1388888889). Square it (0.01929012346) and add it to both sides. Add '0.01929012346' to each side of the equation. -0.2777777778v + 0.01929012346 + v2 = 0.3333333333 + 0.01929012346 Reorder the terms: 0.01929012346 + -0.2777777778v + v2 = 0.3333333333 + 0.01929012346 Combine like terms: 0.3333333333 + 0.01929012346 = 0.35262345676 0.01929012346 + -0.2777777778v + v2 = 0.35262345676 Factor a perfect square on the left side: (v + -0.1388888889)(v + -0.1388888889) = 0.35262345676 Calculate the square root of the right side: 0.593821065 Break this problem into two subproblems by setting (v + -0.1388888889) equal to 0.593821065 and -0.593821065.Subproblem 1
v + -0.1388888889 = 0.593821065 Simplifying v + -0.1388888889 = 0.593821065 Reorder the terms: -0.1388888889 + v = 0.593821065 Solving -0.1388888889 + v = 0.593821065 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '0.1388888889' to each side of the equation. -0.1388888889 + 0.1388888889 + v = 0.593821065 + 0.1388888889 Combine like terms: -0.1388888889 + 0.1388888889 = 0.0000000000 0.0000000000 + v = 0.593821065 + 0.1388888889 v = 0.593821065 + 0.1388888889 Combine like terms: 0.593821065 + 0.1388888889 = 0.7327099539 v = 0.7327099539 Simplifying v = 0.7327099539Subproblem 2
v + -0.1388888889 = -0.593821065 Simplifying v + -0.1388888889 = -0.593821065 Reorder the terms: -0.1388888889 + v = -0.593821065 Solving -0.1388888889 + v = -0.593821065 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '0.1388888889' to each side of the equation. -0.1388888889 + 0.1388888889 + v = -0.593821065 + 0.1388888889 Combine like terms: -0.1388888889 + 0.1388888889 = 0.0000000000 0.0000000000 + v = -0.593821065 + 0.1388888889 v = -0.593821065 + 0.1388888889 Combine like terms: -0.593821065 + 0.1388888889 = -0.4549321761 v = -0.4549321761 Simplifying v = -0.4549321761Solution
The solution to the problem is based on the solutions from the subproblems. v = {0.7327099539, -0.4549321761}
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