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Simplifying 20 = -16t2 + 45t + 6 Reorder the terms: 20 = 6 + 45t + -16t2 Solving 20 = 6 + 45t + -16t2 Solving for variable 't'. Combine like terms: 20 + -6 = 14 14 + -45t + 16t2 = 6 + 45t + -16t2 + -6 + -45t + 16t2 Reorder the terms: 14 + -45t + 16t2 = 6 + -6 + 45t + -45t + -16t2 + 16t2 Combine like terms: 6 + -6 = 0 14 + -45t + 16t2 = 0 + 45t + -45t + -16t2 + 16t2 14 + -45t + 16t2 = 45t + -45t + -16t2 + 16t2 Combine like terms: 45t + -45t = 0 14 + -45t + 16t2 = 0 + -16t2 + 16t2 14 + -45t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 14 + -45t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 0.875 + -2.8125t + t2 = 0 Move the constant term to the right: Add '-0.875' to each side of the equation. 0.875 + -2.8125t + -0.875 + t2 = 0 + -0.875 Reorder the terms: 0.875 + -0.875 + -2.8125t + t2 = 0 + -0.875 Combine like terms: 0.875 + -0.875 = 0.000 0.000 + -2.8125t + t2 = 0 + -0.875 -2.8125t + t2 = 0 + -0.875 Combine like terms: 0 + -0.875 = -0.875 -2.8125t + t2 = -0.875 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 = -0.875 + 1.977539063 Reorder the terms: 1.977539063 + -2.8125t + t2 = -0.875 + 1.977539063 Combine like terms: -0.875 + 1.977539063 = 1.102539063 1.977539063 + -2.8125t + t2 = 1.102539063 Factor a perfect square on the left side: (t + -1.40625)(t + -1.40625) = 1.102539063 Calculate the square root of the right side: 1.050018601 Break this problem into two subproblems by setting (t + -1.40625) equal to 1.050018601 and -1.050018601.Subproblem 1
t + -1.40625 = 1.050018601 Simplifying t + -1.40625 = 1.050018601 Reorder the terms: -1.40625 + t = 1.050018601 Solving -1.40625 + t = 1.050018601 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 = 1.050018601 + 1.40625 Combine like terms: -1.40625 + 1.40625 = 0.00000 0.00000 + t = 1.050018601 + 1.40625 t = 1.050018601 + 1.40625 Combine like terms: 1.050018601 + 1.40625 = 2.456268601 t = 2.456268601 Simplifying t = 2.456268601Subproblem 2
t + -1.40625 = -1.050018601 Simplifying t + -1.40625 = -1.050018601 Reorder the terms: -1.40625 + t = -1.050018601 Solving -1.40625 + t = -1.050018601 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 = -1.050018601 + 1.40625 Combine like terms: -1.40625 + 1.40625 = 0.00000 0.00000 + t = -1.050018601 + 1.40625 t = -1.050018601 + 1.40625 Combine like terms: -1.050018601 + 1.40625 = 0.356231399 t = 0.356231399 Simplifying t = 0.356231399Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.456268601, 0.356231399}
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