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Simplifying 0 = -16t2 + 45t + 2.5 Reorder the terms: 0 = 2.5 + 45t + -16t2 Solving 0 = 2.5 + 45t + -16t2 Solving for variable 't'. Combine like terms: 0 + -2.5 = -2.5 -2.5 + -45t + 16t2 = 2.5 + 45t + -16t2 + -2.5 + -45t + 16t2 Reorder the terms: -2.5 + -45t + 16t2 = 2.5 + -2.5 + 45t + -45t + -16t2 + 16t2 Combine like terms: 2.5 + -2.5 = 0.0 -2.5 + -45t + 16t2 = 0.0 + 45t + -45t + -16t2 + 16t2 -2.5 + -45t + 16t2 = 45t + -45t + -16t2 + 16t2 Combine like terms: 45t + -45t = 0 -2.5 + -45t + 16t2 = 0 + -16t2 + 16t2 -2.5 + -45t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -2.5 + -45t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -0.15625 + -2.8125t + t2 = 0 Move the constant term to the right: Add '0.15625' to each side of the equation. -0.15625 + -2.8125t + 0.15625 + t2 = 0 + 0.15625 Reorder the terms: -0.15625 + 0.15625 + -2.8125t + t2 = 0 + 0.15625 Combine like terms: -0.15625 + 0.15625 = 0.00000 0.00000 + -2.8125t + t2 = 0 + 0.15625 -2.8125t + t2 = 0 + 0.15625 Combine like terms: 0 + 0.15625 = 0.15625 -2.8125t + t2 = 0.15625 The t term is -2.8125t. Take half its coefficient (-1.40625). Square it (1.977539063) and add it to both sides. Add '1.977539063' to each side of the equation. -2.8125t + 1.977539063 + t2 = 0.15625 + 1.977539063 Reorder the terms: 1.977539063 + -2.8125t + t2 = 0.15625 + 1.977539063 Combine like terms: 0.15625 + 1.977539063 = 2.133789063 1.977539063 + -2.8125t + t2 = 2.133789063 Factor a perfect square on the left side: (t + -1.40625)(t + -1.40625) = 2.133789063 Calculate the square root of the right side: 1.460749487 Break this problem into two subproblems by setting (t + -1.40625) equal to 1.460749487 and -1.460749487.Subproblem 1
t + -1.40625 = 1.460749487 Simplifying t + -1.40625 = 1.460749487 Reorder the terms: -1.40625 + t = 1.460749487 Solving -1.40625 + t = 1.460749487 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.40625' to each side of the equation. -1.40625 + 1.40625 + t = 1.460749487 + 1.40625 Combine like terms: -1.40625 + 1.40625 = 0.00000 0.00000 + t = 1.460749487 + 1.40625 t = 1.460749487 + 1.40625 Combine like terms: 1.460749487 + 1.40625 = 2.866999487 t = 2.866999487 Simplifying t = 2.866999487Subproblem 2
t + -1.40625 = -1.460749487 Simplifying t + -1.40625 = -1.460749487 Reorder the terms: -1.40625 + t = -1.460749487 Solving -1.40625 + t = -1.460749487 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.40625' to each side of the equation. -1.40625 + 1.40625 + t = -1.460749487 + 1.40625 Combine like terms: -1.40625 + 1.40625 = 0.00000 0.00000 + t = -1.460749487 + 1.40625 t = -1.460749487 + 1.40625 Combine like terms: -1.460749487 + 1.40625 = -0.054499487 t = -0.054499487 Simplifying t = -0.054499487Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.866999487, -0.054499487}
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