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Simplifying 53 = -16t2 + 75t + 30 Reorder the terms: 53 = 30 + 75t + -16t2 Solving 53 = 30 + 75t + -16t2 Solving for variable 't'. Combine like terms: 53 + -30 = 23 23 + -75t + 16t2 = 30 + 75t + -16t2 + -30 + -75t + 16t2 Reorder the terms: 23 + -75t + 16t2 = 30 + -30 + 75t + -75t + -16t2 + 16t2 Combine like terms: 30 + -30 = 0 23 + -75t + 16t2 = 0 + 75t + -75t + -16t2 + 16t2 23 + -75t + 16t2 = 75t + -75t + -16t2 + 16t2 Combine like terms: 75t + -75t = 0 23 + -75t + 16t2 = 0 + -16t2 + 16t2 23 + -75t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 23 + -75t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 1.4375 + -4.6875t + t2 = 0 Move the constant term to the right: Add '-1.4375' to each side of the equation. 1.4375 + -4.6875t + -1.4375 + t2 = 0 + -1.4375 Reorder the terms: 1.4375 + -1.4375 + -4.6875t + t2 = 0 + -1.4375 Combine like terms: 1.4375 + -1.4375 = 0.0000 0.0000 + -4.6875t + t2 = 0 + -1.4375 -4.6875t + t2 = 0 + -1.4375 Combine like terms: 0 + -1.4375 = -1.4375 -4.6875t + t2 = -1.4375 The t term is -4.6875t. Take half its coefficient (-2.34375). Square it (5.493164063) and add it to both sides. Add '5.493164063' to each side of the equation. -4.6875t + 5.493164063 + t2 = -1.4375 + 5.493164063 Reorder the terms: 5.493164063 + -4.6875t + t2 = -1.4375 + 5.493164063 Combine like terms: -1.4375 + 5.493164063 = 4.055664063 5.493164063 + -4.6875t + t2 = 4.055664063 Factor a perfect square on the left side: (t + -2.34375)(t + -2.34375) = 4.055664063 Calculate the square root of the right side: 2.013867936 Break this problem into two subproblems by setting (t + -2.34375) equal to 2.013867936 and -2.013867936.Subproblem 1
t + -2.34375 = 2.013867936 Simplifying t + -2.34375 = 2.013867936 Reorder the terms: -2.34375 + t = 2.013867936 Solving -2.34375 + t = 2.013867936 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.34375' to each side of the equation. -2.34375 + 2.34375 + t = 2.013867936 + 2.34375 Combine like terms: -2.34375 + 2.34375 = 0.00000 0.00000 + t = 2.013867936 + 2.34375 t = 2.013867936 + 2.34375 Combine like terms: 2.013867936 + 2.34375 = 4.357617936 t = 4.357617936 Simplifying t = 4.357617936Subproblem 2
t + -2.34375 = -2.013867936 Simplifying t + -2.34375 = -2.013867936 Reorder the terms: -2.34375 + t = -2.013867936 Solving -2.34375 + t = -2.013867936 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.34375' to each side of the equation. -2.34375 + 2.34375 + t = -2.013867936 + 2.34375 Combine like terms: -2.34375 + 2.34375 = 0.00000 0.00000 + t = -2.013867936 + 2.34375 t = -2.013867936 + 2.34375 Combine like terms: -2.013867936 + 2.34375 = 0.329882064 t = 0.329882064 Simplifying t = 0.329882064Solution
The solution to the problem is based on the solutions from the subproblems. t = {4.357617936, 0.329882064}
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