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Simplifying v(v + -31.8) + (v + 23.6) = 192.6 Reorder the terms: v(-31.8 + v) + (v + 23.6) = 192.6 (-31.8 * v + v * v) + (v + 23.6) = 192.6 (-31.8v + v2) + (v + 23.6) = 192.6 Reorder the terms: -31.8v + v2 + (23.6 + v) = 192.6 Remove parenthesis around (23.6 + v) -31.8v + v2 + 23.6 + v = 192.6 Reorder the terms: 23.6 + -31.8v + v + v2 = 192.6 Combine like terms: -31.8v + v = -30.8v 23.6 + -30.8v + v2 = 192.6 Solving 23.6 + -30.8v + v2 = 192.6 Solving for variable 'v'. Reorder the terms: 23.6 + -192.6 + -30.8v + v2 = 192.6 + -192.6 Combine like terms: 23.6 + -192.6 = -169 -169 + -30.8v + v2 = 192.6 + -192.6 Combine like terms: 192.6 + -192.6 = 0.0 -169 + -30.8v + v2 = 0.0 Begin completing the square. Move the constant term to the right: Add '169' to each side of the equation. -169 + -30.8v + 169 + v2 = 0.0 + 169 Reorder the terms: -169 + 169 + -30.8v + v2 = 0.0 + 169 Combine like terms: -169 + 169 = 0 0 + -30.8v + v2 = 0.0 + 169 -30.8v + v2 = 0.0 + 169 Combine like terms: 0.0 + 169 = 169 -30.8v + v2 = 169 The v term is -30.8v. Take half its coefficient (-15.4). Square it (237.16) and add it to both sides. Add '237.16' to each side of the equation. -30.8v + 237.16 + v2 = 169 + 237.16 Reorder the terms: 237.16 + -30.8v + v2 = 169 + 237.16 Combine like terms: 169 + 237.16 = 406.16 237.16 + -30.8v + v2 = 406.16 Factor a perfect square on the left side: (v + -15.4)(v + -15.4) = 406.16 Calculate the square root of the right side: 20.153411622 Break this problem into two subproblems by setting (v + -15.4) equal to 20.153411622 and -20.153411622.Subproblem 1
v + -15.4 = 20.153411622 Simplifying v + -15.4 = 20.153411622 Reorder the terms: -15.4 + v = 20.153411622 Solving -15.4 + v = 20.153411622 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '15.4' to each side of the equation. -15.4 + 15.4 + v = 20.153411622 + 15.4 Combine like terms: -15.4 + 15.4 = 0.0 0.0 + v = 20.153411622 + 15.4 v = 20.153411622 + 15.4 Combine like terms: 20.153411622 + 15.4 = 35.553411622 v = 35.553411622 Simplifying v = 35.553411622Subproblem 2
v + -15.4 = -20.153411622 Simplifying v + -15.4 = -20.153411622 Reorder the terms: -15.4 + v = -20.153411622 Solving -15.4 + v = -20.153411622 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '15.4' to each side of the equation. -15.4 + 15.4 + v = -20.153411622 + 15.4 Combine like terms: -15.4 + 15.4 = 0.0 0.0 + v = -20.153411622 + 15.4 v = -20.153411622 + 15.4 Combine like terms: -20.153411622 + 15.4 = -4.753411622 v = -4.753411622 Simplifying v = -4.753411622Solution
The solution to the problem is based on the solutions from the subproblems. v = {35.553411622, -4.753411622}
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