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Simplifying 15 = 0.002(2v2 + 120v + 3000) Reorder the terms: 15 = 0.002(3000 + 120v + 2v2) 15 = (3000 * 0.002 + 120v * 0.002 + 2v2 * 0.002) 15 = (6 + 0.24v + 0.004v2) Solving 15 = 6 + 0.24v + 0.004v2 Solving for variable 'v'. Combine like terms: 15 + -6 = 9 9 + -0.24v + -0.004v2 = 6 + 0.24v + 0.004v2 + -6 + -0.24v + -0.004v2 Reorder the terms: 9 + -0.24v + -0.004v2 = 6 + -6 + 0.24v + -0.24v + 0.004v2 + -0.004v2 Combine like terms: 6 + -6 = 0 9 + -0.24v + -0.004v2 = 0 + 0.24v + -0.24v + 0.004v2 + -0.004v2 9 + -0.24v + -0.004v2 = 0.24v + -0.24v + 0.004v2 + -0.004v2 Combine like terms: 0.24v + -0.24v = 0.00 9 + -0.24v + -0.004v2 = 0.00 + 0.004v2 + -0.004v2 9 + -0.24v + -0.004v2 = 0.004v2 + -0.004v2 Combine like terms: 0.004v2 + -0.004v2 = 0.000 9 + -0.24v + -0.004v2 = 0.000 Begin completing the square. Divide all terms by -0.004 the coefficient of the squared term: Divide each side by '-0.004'. -2250 + 60v + v2 = 0 Move the constant term to the right: Add '2250' to each side of the equation. -2250 + 60v + 2250 + v2 = 0 + 2250 Reorder the terms: -2250 + 2250 + 60v + v2 = 0 + 2250 Combine like terms: -2250 + 2250 = 0 0 + 60v + v2 = 0 + 2250 60v + v2 = 0 + 2250 Combine like terms: 0 + 2250 = 2250 60v + v2 = 2250 The v term is 60v. Take half its coefficient (30). Square it (900) and add it to both sides. Add '900' to each side of the equation. 60v + 900 + v2 = 2250 + 900 Reorder the terms: 900 + 60v + v2 = 2250 + 900 Combine like terms: 2250 + 900 = 3150 900 + 60v + v2 = 3150 Factor a perfect square on the left side: (v + 30)(v + 30) = 3150 Calculate the square root of the right side: 56.124860802 Break this problem into two subproblems by setting (v + 30) equal to 56.124860802 and -56.124860802.Subproblem 1
v + 30 = 56.124860802 Simplifying v + 30 = 56.124860802 Reorder the terms: 30 + v = 56.124860802 Solving 30 + v = 56.124860802 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-30' to each side of the equation. 30 + -30 + v = 56.124860802 + -30 Combine like terms: 30 + -30 = 0 0 + v = 56.124860802 + -30 v = 56.124860802 + -30 Combine like terms: 56.124860802 + -30 = 26.124860802 v = 26.124860802 Simplifying v = 26.124860802Subproblem 2
v + 30 = -56.124860802 Simplifying v + 30 = -56.124860802 Reorder the terms: 30 + v = -56.124860802 Solving 30 + v = -56.124860802 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-30' to each side of the equation. 30 + -30 + v = -56.124860802 + -30 Combine like terms: 30 + -30 = 0 0 + v = -56.124860802 + -30 v = -56.124860802 + -30 Combine like terms: -56.124860802 + -30 = -86.124860802 v = -86.124860802 Simplifying v = -86.124860802Solution
The solution to the problem is based on the solutions from the subproblems. v = {26.124860802, -86.124860802}
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