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Simplifying 7.5 = 4 + 35t + -16t2 Solving 7.5 = 4 + 35t + -16t2 Solving for variable 't'. Combine like terms: 7.5 + -4 = 3.5 3.5 + -35t + 16t2 = 4 + 35t + -16t2 + -4 + -35t + 16t2 Reorder the terms: 3.5 + -35t + 16t2 = 4 + -4 + 35t + -35t + -16t2 + 16t2 Combine like terms: 4 + -4 = 0 3.5 + -35t + 16t2 = 0 + 35t + -35t + -16t2 + 16t2 3.5 + -35t + 16t2 = 35t + -35t + -16t2 + 16t2 Combine like terms: 35t + -35t = 0 3.5 + -35t + 16t2 = 0 + -16t2 + 16t2 3.5 + -35t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 3.5 + -35t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 0.21875 + -2.1875t + t2 = 0 Move the constant term to the right: Add '-0.21875' to each side of the equation. 0.21875 + -2.1875t + -0.21875 + t2 = 0 + -0.21875 Reorder the terms: 0.21875 + -0.21875 + -2.1875t + t2 = 0 + -0.21875 Combine like terms: 0.21875 + -0.21875 = 0.00000 0.00000 + -2.1875t + t2 = 0 + -0.21875 -2.1875t + t2 = 0 + -0.21875 Combine like terms: 0 + -0.21875 = -0.21875 -2.1875t + t2 = -0.21875 The t term is -2.1875t. Take half its coefficient (-1.09375). Square it (1.196289063) and add it to both sides. Add '1.196289063' to each side of the equation. -2.1875t + 1.196289063 + t2 = -0.21875 + 1.196289063 Reorder the terms: 1.196289063 + -2.1875t + t2 = -0.21875 + 1.196289063 Combine like terms: -0.21875 + 1.196289063 = 0.977539063 1.196289063 + -2.1875t + t2 = 0.977539063 Factor a perfect square on the left side: (t + -1.09375)(t + -1.09375) = 0.977539063 Calculate the square root of the right side: 0.988705751 Break this problem into two subproblems by setting (t + -1.09375) equal to 0.988705751 and -0.988705751.Subproblem 1
t + -1.09375 = 0.988705751 Simplifying t + -1.09375 = 0.988705751 Reorder the terms: -1.09375 + t = 0.988705751 Solving -1.09375 + t = 0.988705751 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.09375' to each side of the equation. -1.09375 + 1.09375 + t = 0.988705751 + 1.09375 Combine like terms: -1.09375 + 1.09375 = 0.00000 0.00000 + t = 0.988705751 + 1.09375 t = 0.988705751 + 1.09375 Combine like terms: 0.988705751 + 1.09375 = 2.082455751 t = 2.082455751 Simplifying t = 2.082455751Subproblem 2
t + -1.09375 = -0.988705751 Simplifying t + -1.09375 = -0.988705751 Reorder the terms: -1.09375 + t = -0.988705751 Solving -1.09375 + t = -0.988705751 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.09375' to each side of the equation. -1.09375 + 1.09375 + t = -0.988705751 + 1.09375 Combine like terms: -1.09375 + 1.09375 = 0.00000 0.00000 + t = -0.988705751 + 1.09375 t = -0.988705751 + 1.09375 Combine like terms: -0.988705751 + 1.09375 = 0.105044249 t = 0.105044249 Simplifying t = 0.105044249Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.082455751, 0.105044249}
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