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Simplifying 0.5t = -4.9t2 + 40t + 3 Reorder the terms: 0.5t = 3 + 40t + -4.9t2 Solving 0.5t = 3 + 40t + -4.9t2 Solving for variable 't'. Reorder the terms: -3 + 0.5t + -40t + 4.9t2 = 3 + 40t + -4.9t2 + -3 + -40t + 4.9t2 Combine like terms: 0.5t + -40t = -39.5t -3 + -39.5t + 4.9t2 = 3 + 40t + -4.9t2 + -3 + -40t + 4.9t2 Reorder the terms: -3 + -39.5t + 4.9t2 = 3 + -3 + 40t + -40t + -4.9t2 + 4.9t2 Combine like terms: 3 + -3 = 0 -3 + -39.5t + 4.9t2 = 0 + 40t + -40t + -4.9t2 + 4.9t2 -3 + -39.5t + 4.9t2 = 40t + -40t + -4.9t2 + 4.9t2 Combine like terms: 40t + -40t = 0 -3 + -39.5t + 4.9t2 = 0 + -4.9t2 + 4.9t2 -3 + -39.5t + 4.9t2 = -4.9t2 + 4.9t2 Combine like terms: -4.9t2 + 4.9t2 = 0.0 -3 + -39.5t + 4.9t2 = 0.0 Begin completing the square. Divide all terms by 4.9 the coefficient of the squared term: Divide each side by '4.9'. -0.612244898 + -8.06122449t + t2 = 0 Move the constant term to the right: Add '0.612244898' to each side of the equation. -0.612244898 + -8.06122449t + 0.612244898 + t2 = 0 + 0.612244898 Reorder the terms: -0.612244898 + 0.612244898 + -8.06122449t + t2 = 0 + 0.612244898 Combine like terms: -0.612244898 + 0.612244898 = 0.000000000 0.000000000 + -8.06122449t + t2 = 0 + 0.612244898 -8.06122449t + t2 = 0 + 0.612244898 Combine like terms: 0 + 0.612244898 = 0.612244898 -8.06122449t + t2 = 0.612244898 The t term is -8.06122449t. Take half its coefficient (-4.030612245). Square it (16.24583507) and add it to both sides. Add '16.24583507' to each side of the equation. -8.06122449t + 16.24583507 + t2 = 0.612244898 + 16.24583507 Reorder the terms: 16.24583507 + -8.06122449t + t2 = 0.612244898 + 16.24583507 Combine like terms: 0.612244898 + 16.24583507 = 16.858079968 16.24583507 + -8.06122449t + t2 = 16.858079968 Factor a perfect square on the left side: (t + -4.030612245)(t + -4.030612245) = 16.858079968 Calculate the square root of the right side: 4.105859224 Break this problem into two subproblems by setting (t + -4.030612245) equal to 4.105859224 and -4.105859224.Subproblem 1
t + -4.030612245 = 4.105859224 Simplifying t + -4.030612245 = 4.105859224 Reorder the terms: -4.030612245 + t = 4.105859224 Solving -4.030612245 + t = 4.105859224 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '4.030612245' to each side of the equation. -4.030612245 + 4.030612245 + t = 4.105859224 + 4.030612245 Combine like terms: -4.030612245 + 4.030612245 = 0.000000000 0.000000000 + t = 4.105859224 + 4.030612245 t = 4.105859224 + 4.030612245 Combine like terms: 4.105859224 + 4.030612245 = 8.136471469 t = 8.136471469 Simplifying t = 8.136471469Subproblem 2
t + -4.030612245 = -4.105859224 Simplifying t + -4.030612245 = -4.105859224 Reorder the terms: -4.030612245 + t = -4.105859224 Solving -4.030612245 + t = -4.105859224 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '4.030612245' to each side of the equation. -4.030612245 + 4.030612245 + t = -4.105859224 + 4.030612245 Combine like terms: -4.030612245 + 4.030612245 = 0.000000000 0.000000000 + t = -4.105859224 + 4.030612245 t = -4.105859224 + 4.030612245 Combine like terms: -4.105859224 + 4.030612245 = -0.075246979 t = -0.075246979 Simplifying t = -0.075246979Solution
The solution to the problem is based on the solutions from the subproblems. t = {8.136471469, -0.075246979}
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