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Simplifying 0 = 5 + 45t + -16t2 Solving 0 = 5 + 45t + -16t2 Solving for variable 't'. Combine like terms: 0 + -5 = -5 -5 + -45t + 16t2 = 5 + 45t + -16t2 + -5 + -45t + 16t2 Reorder the terms: -5 + -45t + 16t2 = 5 + -5 + 45t + -45t + -16t2 + 16t2 Combine like terms: 5 + -5 = 0 -5 + -45t + 16t2 = 0 + 45t + -45t + -16t2 + 16t2 -5 + -45t + 16t2 = 45t + -45t + -16t2 + 16t2 Combine like terms: 45t + -45t = 0 -5 + -45t + 16t2 = 0 + -16t2 + 16t2 -5 + -45t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -5 + -45t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -0.3125 + -2.8125t + t2 = 0 Move the constant term to the right: Add '0.3125' to each side of the equation. -0.3125 + -2.8125t + 0.3125 + t2 = 0 + 0.3125 Reorder the terms: -0.3125 + 0.3125 + -2.8125t + t2 = 0 + 0.3125 Combine like terms: -0.3125 + 0.3125 = 0.0000 0.0000 + -2.8125t + t2 = 0 + 0.3125 -2.8125t + t2 = 0 + 0.3125 Combine like terms: 0 + 0.3125 = 0.3125 -2.8125t + t2 = 0.3125 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 = 0.3125 + 1.977539063 Reorder the terms: 1.977539063 + -2.8125t + t2 = 0.3125 + 1.977539063 Combine like terms: 0.3125 + 1.977539063 = 2.290039063 1.977539063 + -2.8125t + t2 = 2.290039063 Factor a perfect square on the left side: (t + -1.40625)(t + -1.40625) = 2.290039063 Calculate the square root of the right side: 1.513287502 Break this problem into two subproblems by setting (t + -1.40625) equal to 1.513287502 and -1.513287502.Subproblem 1
t + -1.40625 = 1.513287502 Simplifying t + -1.40625 = 1.513287502 Reorder the terms: -1.40625 + t = 1.513287502 Solving -1.40625 + t = 1.513287502 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 = 1.513287502 + 1.40625 Combine like terms: -1.40625 + 1.40625 = 0.00000 0.00000 + t = 1.513287502 + 1.40625 t = 1.513287502 + 1.40625 Combine like terms: 1.513287502 + 1.40625 = 2.919537502 t = 2.919537502 Simplifying t = 2.919537502Subproblem 2
t + -1.40625 = -1.513287502 Simplifying t + -1.40625 = -1.513287502 Reorder the terms: -1.40625 + t = -1.513287502 Solving -1.40625 + t = -1.513287502 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 = -1.513287502 + 1.40625 Combine like terms: -1.40625 + 1.40625 = 0.00000 0.00000 + t = -1.513287502 + 1.40625 t = -1.513287502 + 1.40625 Combine like terms: -1.513287502 + 1.40625 = -0.107037502 t = -0.107037502 Simplifying t = -0.107037502Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.919537502, -0.107037502}
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