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Simplifying 3(t) = 67 + -16t2 Solving 3t = 67 + -16t2 Solving for variable 't'. Reorder the terms: -67 + 3t + 16t2 = 67 + -16t2 + -67 + 16t2 Reorder the terms: -67 + 3t + 16t2 = 67 + -67 + -16t2 + 16t2 Combine like terms: 67 + -67 = 0 -67 + 3t + 16t2 = 0 + -16t2 + 16t2 -67 + 3t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -67 + 3t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -4.1875 + 0.1875t + t2 = 0 Move the constant term to the right: Add '4.1875' to each side of the equation. -4.1875 + 0.1875t + 4.1875 + t2 = 0 + 4.1875 Reorder the terms: -4.1875 + 4.1875 + 0.1875t + t2 = 0 + 4.1875 Combine like terms: -4.1875 + 4.1875 = 0.0000 0.0000 + 0.1875t + t2 = 0 + 4.1875 0.1875t + t2 = 0 + 4.1875 Combine like terms: 0 + 4.1875 = 4.1875 0.1875t + t2 = 4.1875 The t term is 0.1875t. Take half its coefficient (0.09375). Square it (0.0087890625) and add it to both sides. Add '0.0087890625' to each side of the equation. 0.1875t + 0.0087890625 + t2 = 4.1875 + 0.0087890625 Reorder the terms: 0.0087890625 + 0.1875t + t2 = 4.1875 + 0.0087890625 Combine like terms: 4.1875 + 0.0087890625 = 4.1962890625 0.0087890625 + 0.1875t + t2 = 4.1962890625 Factor a perfect square on the left side: (t + 0.09375)(t + 0.09375) = 4.1962890625 Calculate the square root of the right side: 2.048484577 Break this problem into two subproblems by setting (t + 0.09375) equal to 2.048484577 and -2.048484577.Subproblem 1
t + 0.09375 = 2.048484577 Simplifying t + 0.09375 = 2.048484577 Reorder the terms: 0.09375 + t = 2.048484577 Solving 0.09375 + t = 2.048484577 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.09375' to each side of the equation. 0.09375 + -0.09375 + t = 2.048484577 + -0.09375 Combine like terms: 0.09375 + -0.09375 = 0.00000 0.00000 + t = 2.048484577 + -0.09375 t = 2.048484577 + -0.09375 Combine like terms: 2.048484577 + -0.09375 = 1.954734577 t = 1.954734577 Simplifying t = 1.954734577Subproblem 2
t + 0.09375 = -2.048484577 Simplifying t + 0.09375 = -2.048484577 Reorder the terms: 0.09375 + t = -2.048484577 Solving 0.09375 + t = -2.048484577 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.09375' to each side of the equation. 0.09375 + -0.09375 + t = -2.048484577 + -0.09375 Combine like terms: 0.09375 + -0.09375 = 0.00000 0.00000 + t = -2.048484577 + -0.09375 t = -2.048484577 + -0.09375 Combine like terms: -2.048484577 + -0.09375 = -2.142234577 t = -2.142234577 Simplifying t = -2.142234577Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.954734577, -2.142234577}
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