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Simplifying 0 = 16t2 + 29t + 6 Reorder the terms: 0 = 6 + 29t + 16t2 Solving 0 = 6 + 29t + 16t2 Solving for variable 't'. Combine like terms: 0 + -6 = -6 -6 + -29t + -16t2 = 6 + 29t + 16t2 + -6 + -29t + -16t2 Reorder the terms: -6 + -29t + -16t2 = 6 + -6 + 29t + -29t + 16t2 + -16t2 Combine like terms: 6 + -6 = 0 -6 + -29t + -16t2 = 0 + 29t + -29t + 16t2 + -16t2 -6 + -29t + -16t2 = 29t + -29t + 16t2 + -16t2 Combine like terms: 29t + -29t = 0 -6 + -29t + -16t2 = 0 + 16t2 + -16t2 -6 + -29t + -16t2 = 16t2 + -16t2 Combine like terms: 16t2 + -16t2 = 0 -6 + -29t + -16t2 = 0 Factor out the Greatest Common Factor (GCF), '-1'. -1(6 + 29t + 16t2) = 0 Ignore the factor -1.Subproblem 1
Set the factor '(6 + 29t + 16t2)' equal to zero and attempt to solve: Simplifying 6 + 29t + 16t2 = 0 Solving 6 + 29t + 16t2 = 0 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 = -0.238201302 t = -0.238201302 Simplifying t = -0.238201302Subproblem 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 = -1.574298698 t = -1.574298698 Simplifying t = -1.574298698Solution
The solution to the problem is based on the solutions from the subproblems. t = {-0.238201302, -1.574298698}Solution
t = {-0.238201302, -1.574298698}
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