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Simplifying 0 = 100 + 10t(0.5) + -4.9t2 Reorder the terms for easier multiplication: 0 = 100 + 10 * 0.5t + -4.9t2 Multiply 10 * 0.5 0 = 100 + 5t + -4.9t2 Solving 0 = 100 + 5t + -4.9t2 Solving for variable 't'. Combine like terms: 0 + -100 = -100 -100 + -5t + 4.9t2 = 100 + 5t + -4.9t2 + -100 + -5t + 4.9t2 Reorder the terms: -100 + -5t + 4.9t2 = 100 + -100 + 5t + -5t + -4.9t2 + 4.9t2 Combine like terms: 100 + -100 = 0 -100 + -5t + 4.9t2 = 0 + 5t + -5t + -4.9t2 + 4.9t2 -100 + -5t + 4.9t2 = 5t + -5t + -4.9t2 + 4.9t2 Combine like terms: 5t + -5t = 0 -100 + -5t + 4.9t2 = 0 + -4.9t2 + 4.9t2 -100 + -5t + 4.9t2 = -4.9t2 + 4.9t2 Combine like terms: -4.9t2 + 4.9t2 = 0.0 -100 + -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'. -20.40816327 + -1.020408163t + t2 = 0 Move the constant term to the right: Add '20.40816327' to each side of the equation. -20.40816327 + -1.020408163t + 20.40816327 + t2 = 0 + 20.40816327 Reorder the terms: -20.40816327 + 20.40816327 + -1.020408163t + t2 = 0 + 20.40816327 Combine like terms: -20.40816327 + 20.40816327 = 0.00000000 0.00000000 + -1.020408163t + t2 = 0 + 20.40816327 -1.020408163t + t2 = 0 + 20.40816327 Combine like terms: 0 + 20.40816327 = 20.40816327 -1.020408163t + t2 = 20.40816327 The t term is -1.020408163t. Take half its coefficient (-0.5102040815). Square it (0.2603082048) and add it to both sides. Add '0.2603082048' to each side of the equation. -1.020408163t + 0.2603082048 + t2 = 20.40816327 + 0.2603082048 Reorder the terms: 0.2603082048 + -1.020408163t + t2 = 20.40816327 + 0.2603082048 Combine like terms: 20.40816327 + 0.2603082048 = 20.6684714748 0.2603082048 + -1.020408163t + t2 = 20.6684714748 Factor a perfect square on the left side: (t + -0.5102040815)(t + -0.5102040815) = 20.6684714748 Calculate the square root of the right side: 4.546259064 Break this problem into two subproblems by setting (t + -0.5102040815) equal to 4.546259064 and -4.546259064.Subproblem 1
t + -0.5102040815 = 4.546259064 Simplifying t + -0.5102040815 = 4.546259064 Reorder the terms: -0.5102040815 + t = 4.546259064 Solving -0.5102040815 + t = 4.546259064 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.5102040815' to each side of the equation. -0.5102040815 + 0.5102040815 + t = 4.546259064 + 0.5102040815 Combine like terms: -0.5102040815 + 0.5102040815 = 0.0000000000 0.0000000000 + t = 4.546259064 + 0.5102040815 t = 4.546259064 + 0.5102040815 Combine like terms: 4.546259064 + 0.5102040815 = 5.0564631455 t = 5.0564631455 Simplifying t = 5.0564631455Subproblem 2
t + -0.5102040815 = -4.546259064 Simplifying t + -0.5102040815 = -4.546259064 Reorder the terms: -0.5102040815 + t = -4.546259064 Solving -0.5102040815 + t = -4.546259064 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.5102040815' to each side of the equation. -0.5102040815 + 0.5102040815 + t = -4.546259064 + 0.5102040815 Combine like terms: -0.5102040815 + 0.5102040815 = 0.0000000000 0.0000000000 + t = -4.546259064 + 0.5102040815 t = -4.546259064 + 0.5102040815 Combine like terms: -4.546259064 + 0.5102040815 = -4.0360549825 t = -4.0360549825 Simplifying t = -4.0360549825Solution
The solution to the problem is based on the solutions from the subproblems. t = {5.0564631455, -4.0360549825}
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