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Simplifying 18v2 + 67v + 24 = 0 Reorder the terms: 24 + 67v + 18v2 = 0 Solving 24 + 67v + 18v2 = 0 Solving for variable 'v'. Begin completing the square. Divide all terms by 18 the coefficient of the squared term: Divide each side by '18'. 1.333333333 + 3.722222222v + v2 = 0 Move the constant term to the right: Add '-1.333333333' to each side of the equation. 1.333333333 + 3.722222222v + -1.333333333 + v2 = 0 + -1.333333333 Reorder the terms: 1.333333333 + -1.333333333 + 3.722222222v + v2 = 0 + -1.333333333 Combine like terms: 1.333333333 + -1.333333333 = 0.000000000 0.000000000 + 3.722222222v + v2 = 0 + -1.333333333 3.722222222v + v2 = 0 + -1.333333333 Combine like terms: 0 + -1.333333333 = -1.333333333 3.722222222v + v2 = -1.333333333 The v term is 3.722222222v. Take half its coefficient (1.861111111). Square it (3.463734567) and add it to both sides. Add '3.463734567' to each side of the equation. 3.722222222v + 3.463734567 + v2 = -1.333333333 + 3.463734567 Reorder the terms: 3.463734567 + 3.722222222v + v2 = -1.333333333 + 3.463734567 Combine like terms: -1.333333333 + 3.463734567 = 2.130401234 3.463734567 + 3.722222222v + v2 = 2.130401234 Factor a perfect square on the left side: (v + 1.861111111)(v + 1.861111111) = 2.130401234 Calculate the square root of the right side: 1.459589406 Break this problem into two subproblems by setting (v + 1.861111111) equal to 1.459589406 and -1.459589406.Subproblem 1
v + 1.861111111 = 1.459589406 Simplifying v + 1.861111111 = 1.459589406 Reorder the terms: 1.861111111 + v = 1.459589406 Solving 1.861111111 + v = 1.459589406 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-1.861111111' to each side of the equation. 1.861111111 + -1.861111111 + v = 1.459589406 + -1.861111111 Combine like terms: 1.861111111 + -1.861111111 = 0.000000000 0.000000000 + v = 1.459589406 + -1.861111111 v = 1.459589406 + -1.861111111 Combine like terms: 1.459589406 + -1.861111111 = -0.401521705 v = -0.401521705 Simplifying v = -0.401521705Subproblem 2
v + 1.861111111 = -1.459589406 Simplifying v + 1.861111111 = -1.459589406 Reorder the terms: 1.861111111 + v = -1.459589406 Solving 1.861111111 + v = -1.459589406 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-1.861111111' to each side of the equation. 1.861111111 + -1.861111111 + v = -1.459589406 + -1.861111111 Combine like terms: 1.861111111 + -1.861111111 = 0.000000000 0.000000000 + v = -1.459589406 + -1.861111111 v = -1.459589406 + -1.861111111 Combine like terms: -1.459589406 + -1.861111111 = -3.320700517 v = -3.320700517 Simplifying v = -3.320700517Solution
The solution to the problem is based on the solutions from the subproblems. v = {-0.401521705, -3.320700517}
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