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Simplifying 65 = -16t2 + 70t + 6 Reorder the terms: 65 = 6 + 70t + -16t2 Solving 65 = 6 + 70t + -16t2 Solving for variable 't'. Combine like terms: 65 + -6 = 59 59 + -70t + 16t2 = 6 + 70t + -16t2 + -6 + -70t + 16t2 Reorder the terms: 59 + -70t + 16t2 = 6 + -6 + 70t + -70t + -16t2 + 16t2 Combine like terms: 6 + -6 = 0 59 + -70t + 16t2 = 0 + 70t + -70t + -16t2 + 16t2 59 + -70t + 16t2 = 70t + -70t + -16t2 + 16t2 Combine like terms: 70t + -70t = 0 59 + -70t + 16t2 = 0 + -16t2 + 16t2 59 + -70t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 59 + -70t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 3.6875 + -4.375t + t2 = 0 Move the constant term to the right: Add '-3.6875' to each side of the equation. 3.6875 + -4.375t + -3.6875 + t2 = 0 + -3.6875 Reorder the terms: 3.6875 + -3.6875 + -4.375t + t2 = 0 + -3.6875 Combine like terms: 3.6875 + -3.6875 = 0.0000 0.0000 + -4.375t + t2 = 0 + -3.6875 -4.375t + t2 = 0 + -3.6875 Combine like terms: 0 + -3.6875 = -3.6875 -4.375t + t2 = -3.6875 The t term is -4.375t. Take half its coefficient (-2.1875). Square it (4.78515625) and add it to both sides. Add '4.78515625' to each side of the equation. -4.375t + 4.78515625 + t2 = -3.6875 + 4.78515625 Reorder the terms: 4.78515625 + -4.375t + t2 = -3.6875 + 4.78515625 Combine like terms: -3.6875 + 4.78515625 = 1.09765625 4.78515625 + -4.375t + t2 = 1.09765625 Factor a perfect square on the left side: (t + -2.1875)(t + -2.1875) = 1.09765625 Calculate the square root of the right side: 1.047690913 Break this problem into two subproblems by setting (t + -2.1875) equal to 1.047690913 and -1.047690913.Subproblem 1
t + -2.1875 = 1.047690913 Simplifying t + -2.1875 = 1.047690913 Reorder the terms: -2.1875 + t = 1.047690913 Solving -2.1875 + t = 1.047690913 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.1875' to each side of the equation. -2.1875 + 2.1875 + t = 1.047690913 + 2.1875 Combine like terms: -2.1875 + 2.1875 = 0.0000 0.0000 + t = 1.047690913 + 2.1875 t = 1.047690913 + 2.1875 Combine like terms: 1.047690913 + 2.1875 = 3.235190913 t = 3.235190913 Simplifying t = 3.235190913Subproblem 2
t + -2.1875 = -1.047690913 Simplifying t + -2.1875 = -1.047690913 Reorder the terms: -2.1875 + t = -1.047690913 Solving -2.1875 + t = -1.047690913 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.1875' to each side of the equation. -2.1875 + 2.1875 + t = -1.047690913 + 2.1875 Combine like terms: -2.1875 + 2.1875 = 0.0000 0.0000 + t = -1.047690913 + 2.1875 t = -1.047690913 + 2.1875 Combine like terms: -1.047690913 + 2.1875 = 1.139809087 t = 1.139809087 Simplifying t = 1.139809087Solution
The solution to the problem is based on the solutions from the subproblems. t = {3.235190913, 1.139809087}
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