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Simplifying 3 = -16t2 + 29t + 6 Reorder the terms: 3 = 6 + 29t + -16t2 Solving 3 = 6 + 29t + -16t2 Solving for variable 't'. Combine like terms: 3 + -6 = -3 -3 + -29t + 16t2 = 6 + 29t + -16t2 + -6 + -29t + 16t2 Reorder the terms: -3 + -29t + 16t2 = 6 + -6 + 29t + -29t + -16t2 + 16t2 Combine like terms: 6 + -6 = 0 -3 + -29t + 16t2 = 0 + 29t + -29t + -16t2 + 16t2 -3 + -29t + 16t2 = 29t + -29t + -16t2 + 16t2 Combine like terms: 29t + -29t = 0 -3 + -29t + 16t2 = 0 + -16t2 + 16t2 -3 + -29t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -3 + -29t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -0.1875 + -1.8125t + t2 = 0 Move the constant term to the right: Add '0.1875' to each side of the equation. -0.1875 + -1.8125t + 0.1875 + t2 = 0 + 0.1875 Reorder the terms: -0.1875 + 0.1875 + -1.8125t + t2 = 0 + 0.1875 Combine like terms: -0.1875 + 0.1875 = 0.0000 0.0000 + -1.8125t + t2 = 0 + 0.1875 -1.8125t + t2 = 0 + 0.1875 Combine like terms: 0 + 0.1875 = 0.1875 -1.8125t + t2 = 0.1875 The t term is -1.8125t. Take half its coefficient (-0.90625). Square it (0.8212890625) and add it to both sides. Add '0.8212890625' to each side of the equation. -1.8125t + 0.8212890625 + t2 = 0.1875 + 0.8212890625 Reorder the terms: 0.8212890625 + -1.8125t + t2 = 0.1875 + 0.8212890625 Combine like terms: 0.1875 + 0.8212890625 = 1.0087890625 0.8212890625 + -1.8125t + t2 = 1.0087890625 Factor a perfect square on the left side: (t + -0.90625)(t + -0.90625) = 1.0087890625 Calculate the square root of the right side: 1.004384918 Break this problem into two subproblems by setting (t + -0.90625) equal to 1.004384918 and -1.004384918.Subproblem 1
t + -0.90625 = 1.004384918 Simplifying t + -0.90625 = 1.004384918 Reorder the terms: -0.90625 + t = 1.004384918 Solving -0.90625 + t = 1.004384918 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.90625' to each side of the equation. -0.90625 + 0.90625 + t = 1.004384918 + 0.90625 Combine like terms: -0.90625 + 0.90625 = 0.00000 0.00000 + t = 1.004384918 + 0.90625 t = 1.004384918 + 0.90625 Combine like terms: 1.004384918 + 0.90625 = 1.910634918 t = 1.910634918 Simplifying t = 1.910634918Subproblem 2
t + -0.90625 = -1.004384918 Simplifying t + -0.90625 = -1.004384918 Reorder the terms: -0.90625 + t = -1.004384918 Solving -0.90625 + t = -1.004384918 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.90625' to each side of the equation. -0.90625 + 0.90625 + t = -1.004384918 + 0.90625 Combine like terms: -0.90625 + 0.90625 = 0.00000 0.00000 + t = -1.004384918 + 0.90625 t = -1.004384918 + 0.90625 Combine like terms: -1.004384918 + 0.90625 = -0.098134918 t = -0.098134918 Simplifying t = -0.098134918Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.910634918, -0.098134918}
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