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Simplifying 2v2 + 4v + -14 = v + 1v + 1 Reorder the terms: -14 + 4v + 2v2 = v + 1v + 1 Reorder the terms: -14 + 4v + 2v2 = 1 + v + 1v Combine like terms: v + 1v = 2v -14 + 4v + 2v2 = 1 + 2v Solving -14 + 4v + 2v2 = 1 + 2v Solving for variable 'v'. Reorder the terms: -14 + -1 + 4v + -2v + 2v2 = 1 + 2v + -1 + -2v Combine like terms: -14 + -1 = -15 -15 + 4v + -2v + 2v2 = 1 + 2v + -1 + -2v Combine like terms: 4v + -2v = 2v -15 + 2v + 2v2 = 1 + 2v + -1 + -2v Reorder the terms: -15 + 2v + 2v2 = 1 + -1 + 2v + -2v Combine like terms: 1 + -1 = 0 -15 + 2v + 2v2 = 0 + 2v + -2v -15 + 2v + 2v2 = 2v + -2v Combine like terms: 2v + -2v = 0 -15 + 2v + 2v2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -7.5 + v + v2 = 0 Move the constant term to the right: Add '7.5' to each side of the equation. -7.5 + v + 7.5 + v2 = 0 + 7.5 Reorder the terms: -7.5 + 7.5 + v + v2 = 0 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + v + v2 = 0 + 7.5 v + v2 = 0 + 7.5 Combine like terms: 0 + 7.5 = 7.5 v + v2 = 7.5 The v term is v. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. + 0.25 + v2 = 7.5 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + v2 = 7.5 + 0.25 Combine like terms: 7.5 + 0.25 = 7.75 1.25 + v2 = 7.75 Factor a perfect square on the left side: (v + 0.5)(v + 0.5) = 7.75 Calculate the square root of the right side: 2.783882181 Break this problem into two subproblems by setting (v + 0.5) equal to 2.783882181 and -2.783882181.Subproblem 1
v + 0.5 = 2.783882181 Simplifying v + 0.5 = 2.783882181 Reorder the terms: 0.5 + v = 2.783882181 Solving 0.5 + v = 2.783882181 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + v = 2.783882181 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + v = 2.783882181 + -0.5 v = 2.783882181 + -0.5 Combine like terms: 2.783882181 + -0.5 = 2.283882181 v = 2.283882181 Simplifying v = 2.283882181Subproblem 2
v + 0.5 = -2.783882181 Simplifying v + 0.5 = -2.783882181 Reorder the terms: 0.5 + v = -2.783882181 Solving 0.5 + v = -2.783882181 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + v = -2.783882181 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + v = -2.783882181 + -0.5 v = -2.783882181 + -0.5 Combine like terms: -2.783882181 + -0.5 = -3.283882181 v = -3.283882181 Simplifying v = -3.283882181Solution
The solution to the problem is based on the solutions from the subproblems. v = {2.283882181, -3.283882181}
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