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Simplifying 2v2 + -31v + 18 = 8v2 + -2 + 2v Reorder the terms: 18 + -31v + 2v2 = 8v2 + -2 + 2v Reorder the terms: 18 + -31v + 2v2 = -2 + 2v + 8v2 Solving 18 + -31v + 2v2 = -2 + 2v + 8v2 Solving for variable 'v'. Reorder the terms: 18 + 2 + -31v + -2v + 2v2 + -8v2 = -2 + 2v + 8v2 + 2 + -2v + -8v2 Combine like terms: 18 + 2 = 20 20 + -31v + -2v + 2v2 + -8v2 = -2 + 2v + 8v2 + 2 + -2v + -8v2 Combine like terms: -31v + -2v = -33v 20 + -33v + 2v2 + -8v2 = -2 + 2v + 8v2 + 2 + -2v + -8v2 Combine like terms: 2v2 + -8v2 = -6v2 20 + -33v + -6v2 = -2 + 2v + 8v2 + 2 + -2v + -8v2 Reorder the terms: 20 + -33v + -6v2 = -2 + 2 + 2v + -2v + 8v2 + -8v2 Combine like terms: -2 + 2 = 0 20 + -33v + -6v2 = 0 + 2v + -2v + 8v2 + -8v2 20 + -33v + -6v2 = 2v + -2v + 8v2 + -8v2 Combine like terms: 2v + -2v = 0 20 + -33v + -6v2 = 0 + 8v2 + -8v2 20 + -33v + -6v2 = 8v2 + -8v2 Combine like terms: 8v2 + -8v2 = 0 20 + -33v + -6v2 = 0 Begin completing the square. Divide all terms by -6 the coefficient of the squared term: Divide each side by '-6'. -3.333333333 + 5.5v + v2 = 0 Move the constant term to the right: Add '3.333333333' to each side of the equation. -3.333333333 + 5.5v + 3.333333333 + v2 = 0 + 3.333333333 Reorder the terms: -3.333333333 + 3.333333333 + 5.5v + v2 = 0 + 3.333333333 Combine like terms: -3.333333333 + 3.333333333 = 0.000000000 0.000000000 + 5.5v + v2 = 0 + 3.333333333 5.5v + v2 = 0 + 3.333333333 Combine like terms: 0 + 3.333333333 = 3.333333333 5.5v + v2 = 3.333333333 The v term is 5.5v. Take half its coefficient (2.75). Square it (7.5625) and add it to both sides. Add '7.5625' to each side of the equation. 5.5v + 7.5625 + v2 = 3.333333333 + 7.5625 Reorder the terms: 7.5625 + 5.5v + v2 = 3.333333333 + 7.5625 Combine like terms: 3.333333333 + 7.5625 = 10.895833333 7.5625 + 5.5v + v2 = 10.895833333 Factor a perfect square on the left side: (v + 2.75)(v + 2.75) = 10.895833333 Calculate the square root of the right side: 3.30088372 Break this problem into two subproblems by setting (v + 2.75) equal to 3.30088372 and -3.30088372.Subproblem 1
v + 2.75 = 3.30088372 Simplifying v + 2.75 = 3.30088372 Reorder the terms: 2.75 + v = 3.30088372 Solving 2.75 + v = 3.30088372 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-2.75' to each side of the equation. 2.75 + -2.75 + v = 3.30088372 + -2.75 Combine like terms: 2.75 + -2.75 = 0.00 0.00 + v = 3.30088372 + -2.75 v = 3.30088372 + -2.75 Combine like terms: 3.30088372 + -2.75 = 0.55088372 v = 0.55088372 Simplifying v = 0.55088372Subproblem 2
v + 2.75 = -3.30088372 Simplifying v + 2.75 = -3.30088372 Reorder the terms: 2.75 + v = -3.30088372 Solving 2.75 + v = -3.30088372 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-2.75' to each side of the equation. 2.75 + -2.75 + v = -3.30088372 + -2.75 Combine like terms: 2.75 + -2.75 = 0.00 0.00 + v = -3.30088372 + -2.75 v = -3.30088372 + -2.75 Combine like terms: -3.30088372 + -2.75 = -6.05088372 v = -6.05088372 Simplifying v = -6.05088372Solution
The solution to the problem is based on the solutions from the subproblems. v = {0.55088372, -6.05088372}
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