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Simplifying (7v2 + -4v + -6) + (5v2 + -3v + 2) = 0 Reorder the terms: (-6 + -4v + 7v2) + (5v2 + -3v + 2) = 0 Remove parenthesis around (-6 + -4v + 7v2) -6 + -4v + 7v2 + (5v2 + -3v + 2) = 0 Reorder the terms: -6 + -4v + 7v2 + (2 + -3v + 5v2) = 0 Remove parenthesis around (2 + -3v + 5v2) -6 + -4v + 7v2 + 2 + -3v + 5v2 = 0 Reorder the terms: -6 + 2 + -4v + -3v + 7v2 + 5v2 = 0 Combine like terms: -6 + 2 = -4 -4 + -4v + -3v + 7v2 + 5v2 = 0 Combine like terms: -4v + -3v = -7v -4 + -7v + 7v2 + 5v2 = 0 Combine like terms: 7v2 + 5v2 = 12v2 -4 + -7v + 12v2 = 0 Solving -4 + -7v + 12v2 = 0 Solving for variable 'v'. Begin completing the square. Divide all terms by 12 the coefficient of the squared term: Divide each side by '12'. -0.3333333333 + -0.5833333333v + v2 = 0 Move the constant term to the right: Add '0.3333333333' to each side of the equation. -0.3333333333 + -0.5833333333v + 0.3333333333 + v2 = 0 + 0.3333333333 Reorder the terms: -0.3333333333 + 0.3333333333 + -0.5833333333v + v2 = 0 + 0.3333333333 Combine like terms: -0.3333333333 + 0.3333333333 = 0.0000000000 0.0000000000 + -0.5833333333v + v2 = 0 + 0.3333333333 -0.5833333333v + v2 = 0 + 0.3333333333 Combine like terms: 0 + 0.3333333333 = 0.3333333333 -0.5833333333v + v2 = 0.3333333333 The v term is -0.5833333333v. Take half its coefficient (-0.2916666667). Square it (0.08506944446) and add it to both sides. Add '0.08506944446' to each side of the equation. -0.5833333333v + 0.08506944446 + v2 = 0.3333333333 + 0.08506944446 Reorder the terms: 0.08506944446 + -0.5833333333v + v2 = 0.3333333333 + 0.08506944446 Combine like terms: 0.3333333333 + 0.08506944446 = 0.41840277776 0.08506944446 + -0.5833333333v + v2 = 0.41840277776 Factor a perfect square on the left side: (v + -0.2916666667)(v + -0.2916666667) = 0.41840277776 Calculate the square root of the right side: 0.646840612 Break this problem into two subproblems by setting (v + -0.2916666667) equal to 0.646840612 and -0.646840612.Subproblem 1
v + -0.2916666667 = 0.646840612 Simplifying v + -0.2916666667 = 0.646840612 Reorder the terms: -0.2916666667 + v = 0.646840612 Solving -0.2916666667 + v = 0.646840612 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '0.2916666667' to each side of the equation. -0.2916666667 + 0.2916666667 + v = 0.646840612 + 0.2916666667 Combine like terms: -0.2916666667 + 0.2916666667 = 0.0000000000 0.0000000000 + v = 0.646840612 + 0.2916666667 v = 0.646840612 + 0.2916666667 Combine like terms: 0.646840612 + 0.2916666667 = 0.9385072787 v = 0.9385072787 Simplifying v = 0.9385072787Subproblem 2
v + -0.2916666667 = -0.646840612 Simplifying v + -0.2916666667 = -0.646840612 Reorder the terms: -0.2916666667 + v = -0.646840612 Solving -0.2916666667 + v = -0.646840612 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '0.2916666667' to each side of the equation. -0.2916666667 + 0.2916666667 + v = -0.646840612 + 0.2916666667 Combine like terms: -0.2916666667 + 0.2916666667 = 0.0000000000 0.0000000000 + v = -0.646840612 + 0.2916666667 v = -0.646840612 + 0.2916666667 Combine like terms: -0.646840612 + 0.2916666667 = -0.3551739453 v = -0.3551739453 Simplifying v = -0.3551739453Solution
The solution to the problem is based on the solutions from the subproblems. v = {0.9385072787, -0.3551739453}
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