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Simplifying 2v2 + 12v + 2 = (v + 3) Reorder the terms: 2 + 12v + 2v2 = (v + 3) Reorder the terms: 2 + 12v + 2v2 = (3 + v) Remove parenthesis around (3 + v) 2 + 12v + 2v2 = 3 + v Solving 2 + 12v + 2v2 = 3 + v Solving for variable 'v'. Reorder the terms: 2 + -3 + 12v + -1v + 2v2 = 3 + v + -3 + -1v Combine like terms: 2 + -3 = -1 -1 + 12v + -1v + 2v2 = 3 + v + -3 + -1v Combine like terms: 12v + -1v = 11v -1 + 11v + 2v2 = 3 + v + -3 + -1v Reorder the terms: -1 + 11v + 2v2 = 3 + -3 + v + -1v Combine like terms: 3 + -3 = 0 -1 + 11v + 2v2 = 0 + v + -1v -1 + 11v + 2v2 = v + -1v Combine like terms: v + -1v = 0 -1 + 11v + 2v2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -0.5 + 5.5v + v2 = 0 Move the constant term to the right: Add '0.5' to each side of the equation. -0.5 + 5.5v + 0.5 + v2 = 0 + 0.5 Reorder the terms: -0.5 + 0.5 + 5.5v + v2 = 0 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + 5.5v + v2 = 0 + 0.5 5.5v + v2 = 0 + 0.5 Combine like terms: 0 + 0.5 = 0.5 5.5v + v2 = 0.5 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 = 0.5 + 7.5625 Reorder the terms: 7.5625 + 5.5v + v2 = 0.5 + 7.5625 Combine like terms: 0.5 + 7.5625 = 8.0625 7.5625 + 5.5v + v2 = 8.0625 Factor a perfect square on the left side: (v + 2.75)(v + 2.75) = 8.0625 Calculate the square root of the right side: 2.839454173 Break this problem into two subproblems by setting (v + 2.75) equal to 2.839454173 and -2.839454173.Subproblem 1
v + 2.75 = 2.839454173 Simplifying v + 2.75 = 2.839454173 Reorder the terms: 2.75 + v = 2.839454173 Solving 2.75 + v = 2.839454173 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 = 2.839454173 + -2.75 Combine like terms: 2.75 + -2.75 = 0.00 0.00 + v = 2.839454173 + -2.75 v = 2.839454173 + -2.75 Combine like terms: 2.839454173 + -2.75 = 0.089454173 v = 0.089454173 Simplifying v = 0.089454173Subproblem 2
v + 2.75 = -2.839454173 Simplifying v + 2.75 = -2.839454173 Reorder the terms: 2.75 + v = -2.839454173 Solving 2.75 + v = -2.839454173 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 = -2.839454173 + -2.75 Combine like terms: 2.75 + -2.75 = 0.00 0.00 + v = -2.839454173 + -2.75 v = -2.839454173 + -2.75 Combine like terms: -2.839454173 + -2.75 = -5.589454173 v = -5.589454173 Simplifying v = -5.589454173Solution
The solution to the problem is based on the solutions from the subproblems. v = {0.089454173, -5.589454173}
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