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Simplifying -16t2 + 32t + -3t + -6 = 0 Reorder the terms: -6 + 32t + -3t + -16t2 = 0 Combine like terms: 32t + -3t = 29t -6 + 29t + -16t2 = 0 Solving -6 + 29t + -16t2 = 0 Solving for variable 't'. Begin completing the square. Divide all terms by -16 the coefficient of the squared term: Divide each side by '-16'. 0.375 + -1.8125t + t2 = 0 Move the constant term to the right: Add '-0.375' to each side of the equation. 0.375 + -1.8125t + -0.375 + t2 = 0 + -0.375 Reorder the terms: 0.375 + -0.375 + -1.8125t + t2 = 0 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + -1.8125t + t2 = 0 + -0.375 -1.8125t + t2 = 0 + -0.375 Combine like terms: 0 + -0.375 = -0.375 -1.8125t + t2 = -0.375 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 = -0.375 + 0.8212890625 Reorder the terms: 0.8212890625 + -1.8125t + t2 = -0.375 + 0.8212890625 Combine like terms: -0.375 + 0.8212890625 = 0.4462890625 0.8212890625 + -1.8125t + t2 = 0.4462890625 Factor a perfect square on the left side: (t + -0.90625)(t + -0.90625) = 0.4462890625 Calculate the square root of the right side: 0.668048698 Break this problem into two subproblems by setting (t + -0.90625) equal to 0.668048698 and -0.668048698.Subproblem 1
t + -0.90625 = 0.668048698 Simplifying t + -0.90625 = 0.668048698 Reorder the terms: -0.90625 + t = 0.668048698 Solving -0.90625 + t = 0.668048698 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 = 0.668048698 + 0.90625 Combine like terms: -0.90625 + 0.90625 = 0.00000 0.00000 + t = 0.668048698 + 0.90625 t = 0.668048698 + 0.90625 Combine like terms: 0.668048698 + 0.90625 = 1.574298698 t = 1.574298698 Simplifying t = 1.574298698Subproblem 2
t + -0.90625 = -0.668048698 Simplifying t + -0.90625 = -0.668048698 Reorder the terms: -0.90625 + t = -0.668048698 Solving -0.90625 + t = -0.668048698 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 = -0.668048698 + 0.90625 Combine like terms: -0.90625 + 0.90625 = 0.00000 0.00000 + t = -0.668048698 + 0.90625 t = -0.668048698 + 0.90625 Combine like terms: -0.668048698 + 0.90625 = 0.238201302 t = 0.238201302 Simplifying t = 0.238201302Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.574298698, 0.238201302}
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