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Simplifying 0 = v2 + 0.4v + -450000 Reorder the terms: 0 = -450000 + 0.4v + v2 Solving 0 = -450000 + 0.4v + v2 Solving for variable 'v'. Combine like terms: 0 + 450000 = 450000 450000 + -0.4v + -1v2 = -450000 + 0.4v + v2 + 450000 + -0.4v + -1v2 Reorder the terms: 450000 + -0.4v + -1v2 = -450000 + 450000 + 0.4v + -0.4v + v2 + -1v2 Combine like terms: -450000 + 450000 = 0 450000 + -0.4v + -1v2 = 0 + 0.4v + -0.4v + v2 + -1v2 450000 + -0.4v + -1v2 = 0.4v + -0.4v + v2 + -1v2 Combine like terms: 0.4v + -0.4v = 0.0 450000 + -0.4v + -1v2 = 0.0 + v2 + -1v2 450000 + -0.4v + -1v2 = v2 + -1v2 Combine like terms: v2 + -1v2 = 0 450000 + -0.4v + -1v2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -450000 + 0.4v + v2 = 0 Move the constant term to the right: Add '450000' to each side of the equation. -450000 + 0.4v + 450000 + v2 = 0 + 450000 Reorder the terms: -450000 + 450000 + 0.4v + v2 = 0 + 450000 Combine like terms: -450000 + 450000 = 0 0 + 0.4v + v2 = 0 + 450000 0.4v + v2 = 0 + 450000 Combine like terms: 0 + 450000 = 450000 0.4v + v2 = 450000 The v term is 0.4v. Take half its coefficient (0.2). Square it (0.04) and add it to both sides. Add '0.04' to each side of the equation. 0.4v + 0.04 + v2 = 450000 + 0.04 Reorder the terms: 0.04 + 0.4v + v2 = 450000 + 0.04 Combine like terms: 450000 + 0.04 = 450000.04 0.04 + 0.4v + v2 = 450000.04 Factor a perfect square on the left side: (v + 0.2)(v + 0.2) = 450000.04 Calculate the square root of the right side: 670.820423064 Break this problem into two subproblems by setting (v + 0.2) equal to 670.820423064 and -670.820423064.Subproblem 1
v + 0.2 = 670.820423064 Simplifying v + 0.2 = 670.820423064 Reorder the terms: 0.2 + v = 670.820423064 Solving 0.2 + v = 670.820423064 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-0.2' to each side of the equation. 0.2 + -0.2 + v = 670.820423064 + -0.2 Combine like terms: 0.2 + -0.2 = 0.0 0.0 + v = 670.820423064 + -0.2 v = 670.820423064 + -0.2 Combine like terms: 670.820423064 + -0.2 = 670.620423064 v = 670.620423064 Simplifying v = 670.620423064Subproblem 2
v + 0.2 = -670.820423064 Simplifying v + 0.2 = -670.820423064 Reorder the terms: 0.2 + v = -670.820423064 Solving 0.2 + v = -670.820423064 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-0.2' to each side of the equation. 0.2 + -0.2 + v = -670.820423064 + -0.2 Combine like terms: 0.2 + -0.2 = 0.0 0.0 + v = -670.820423064 + -0.2 v = -670.820423064 + -0.2 Combine like terms: -670.820423064 + -0.2 = -671.020423064 v = -671.020423064 Simplifying v = -671.020423064Solution
The solution to the problem is based on the solutions from the subproblems. v = {670.620423064, -671.020423064}
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