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Simplifying 0 = 2v2 + 1v + -47 Reorder the terms: 0 = -47 + 1v + 2v2 Solving 0 = -47 + 1v + 2v2 Solving for variable 'v'. Combine like terms: 0 + 47 = 47 47 + -1v + -2v2 = -47 + 1v + 2v2 + 47 + -1v + -2v2 Reorder the terms: 47 + -1v + -2v2 = -47 + 47 + 1v + -1v + 2v2 + -2v2 Combine like terms: -47 + 47 = 0 47 + -1v + -2v2 = 0 + 1v + -1v + 2v2 + -2v2 47 + -1v + -2v2 = 1v + -1v + 2v2 + -2v2 Combine like terms: 1v + -1v = 0 47 + -1v + -2v2 = 0 + 2v2 + -2v2 47 + -1v + -2v2 = 2v2 + -2v2 Combine like terms: 2v2 + -2v2 = 0 47 + -1v + -2v2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -23.5 + 0.5v + v2 = 0 Move the constant term to the right: Add '23.5' to each side of the equation. -23.5 + 0.5v + 23.5 + v2 = 0 + 23.5 Reorder the terms: -23.5 + 23.5 + 0.5v + v2 = 0 + 23.5 Combine like terms: -23.5 + 23.5 = 0.0 0.0 + 0.5v + v2 = 0 + 23.5 0.5v + v2 = 0 + 23.5 Combine like terms: 0 + 23.5 = 23.5 0.5v + v2 = 23.5 The v term is 0.5v. Take half its coefficient (0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. 0.5v + 0.0625 + v2 = 23.5 + 0.0625 Reorder the terms: 0.0625 + 0.5v + v2 = 23.5 + 0.0625 Combine like terms: 23.5 + 0.0625 = 23.5625 0.0625 + 0.5v + v2 = 23.5625 Factor a perfect square on the left side: (v + 0.25)(v + 0.25) = 23.5625 Calculate the square root of the right side: 4.85412196 Break this problem into two subproblems by setting (v + 0.25) equal to 4.85412196 and -4.85412196.Subproblem 1
v + 0.25 = 4.85412196 Simplifying v + 0.25 = 4.85412196 Reorder the terms: 0.25 + v = 4.85412196 Solving 0.25 + v = 4.85412196 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + v = 4.85412196 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + v = 4.85412196 + -0.25 v = 4.85412196 + -0.25 Combine like terms: 4.85412196 + -0.25 = 4.60412196 v = 4.60412196 Simplifying v = 4.60412196Subproblem 2
v + 0.25 = -4.85412196 Simplifying v + 0.25 = -4.85412196 Reorder the terms: 0.25 + v = -4.85412196 Solving 0.25 + v = -4.85412196 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + v = -4.85412196 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + v = -4.85412196 + -0.25 v = -4.85412196 + -0.25 Combine like terms: -4.85412196 + -0.25 = -5.10412196 v = -5.10412196 Simplifying v = -5.10412196Solution
The solution to the problem is based on the solutions from the subproblems. v = {4.60412196, -5.10412196}
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