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Simplifying (2v2 + 4v + 7) + -1(6v2 + v + 3) = 0 Reorder the terms: (7 + 4v + 2v2) + -1(6v2 + v + 3) = 0 Remove parenthesis around (7 + 4v + 2v2) 7 + 4v + 2v2 + -1(6v2 + v + 3) = 0 Reorder the terms: 7 + 4v + 2v2 + -1(3 + v + 6v2) = 0 7 + 4v + 2v2 + (3 * -1 + v * -1 + 6v2 * -1) = 0 7 + 4v + 2v2 + (-3 + -1v + -6v2) = 0 Reorder the terms: 7 + -3 + 4v + -1v + 2v2 + -6v2 = 0 Combine like terms: 7 + -3 = 4 4 + 4v + -1v + 2v2 + -6v2 = 0 Combine like terms: 4v + -1v = 3v 4 + 3v + 2v2 + -6v2 = 0 Combine like terms: 2v2 + -6v2 = -4v2 4 + 3v + -4v2 = 0 Solving 4 + 3v + -4v2 = 0 Solving for variable 'v'. Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. -1 + -0.75v + v2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + -0.75v + 1 + v2 = 0 + 1 Reorder the terms: -1 + 1 + -0.75v + v2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + -0.75v + v2 = 0 + 1 -0.75v + v2 = 0 + 1 Combine like terms: 0 + 1 = 1 -0.75v + v2 = 1 The v term is -0.75v. Take half its coefficient (-0.375). Square it (0.140625) and add it to both sides. Add '0.140625' to each side of the equation. -0.75v + 0.140625 + v2 = 1 + 0.140625 Reorder the terms: 0.140625 + -0.75v + v2 = 1 + 0.140625 Combine like terms: 1 + 0.140625 = 1.140625 0.140625 + -0.75v + v2 = 1.140625 Factor a perfect square on the left side: (v + -0.375)(v + -0.375) = 1.140625 Calculate the square root of the right side: 1.068000468 Break this problem into two subproblems by setting (v + -0.375) equal to 1.068000468 and -1.068000468.Subproblem 1
v + -0.375 = 1.068000468 Simplifying v + -0.375 = 1.068000468 Reorder the terms: -0.375 + v = 1.068000468 Solving -0.375 + v = 1.068000468 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '0.375' to each side of the equation. -0.375 + 0.375 + v = 1.068000468 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + v = 1.068000468 + 0.375 v = 1.068000468 + 0.375 Combine like terms: 1.068000468 + 0.375 = 1.443000468 v = 1.443000468 Simplifying v = 1.443000468Subproblem 2
v + -0.375 = -1.068000468 Simplifying v + -0.375 = -1.068000468 Reorder the terms: -0.375 + v = -1.068000468 Solving -0.375 + v = -1.068000468 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '0.375' to each side of the equation. -0.375 + 0.375 + v = -1.068000468 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + v = -1.068000468 + 0.375 v = -1.068000468 + 0.375 Combine like terms: -1.068000468 + 0.375 = -0.693000468 v = -0.693000468 Simplifying v = -0.693000468Solution
The solution to the problem is based on the solutions from the subproblems. v = {1.443000468, -0.693000468}
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