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Simplifying 3.0 = 16t2 + 40t + 4 Reorder the terms: 3.0 = 4 + 40t + 16t2 Solving 3.0 = 4 + 40t + 16t2 Solving for variable 't'. Combine like terms: 3.0 + -4 = -1 -1 + -40t + -16t2 = 4 + 40t + 16t2 + -4 + -40t + -16t2 Reorder the terms: -1 + -40t + -16t2 = 4 + -4 + 40t + -40t + 16t2 + -16t2 Combine like terms: 4 + -4 = 0 -1 + -40t + -16t2 = 0 + 40t + -40t + 16t2 + -16t2 -1 + -40t + -16t2 = 40t + -40t + 16t2 + -16t2 Combine like terms: 40t + -40t = 0 -1 + -40t + -16t2 = 0 + 16t2 + -16t2 -1 + -40t + -16t2 = 16t2 + -16t2 Combine like terms: 16t2 + -16t2 = 0 -1 + -40t + -16t2 = 0 Factor out the Greatest Common Factor (GCF), '-1'. -1(1 + 40t + 16t2) = 0 Ignore the factor -1.Subproblem 1
Set the factor '(1 + 40t + 16t2)' equal to zero and attempt to solve: Simplifying 1 + 40t + 16t2 = 0 Solving 1 + 40t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 0.0625 + 2.5t + t2 = 0 Move the constant term to the right: Add '-0.0625' to each side of the equation. 0.0625 + 2.5t + -0.0625 + t2 = 0 + -0.0625 Reorder the terms: 0.0625 + -0.0625 + 2.5t + t2 = 0 + -0.0625 Combine like terms: 0.0625 + -0.0625 = 0.0000 0.0000 + 2.5t + t2 = 0 + -0.0625 2.5t + t2 = 0 + -0.0625 Combine like terms: 0 + -0.0625 = -0.0625 2.5t + t2 = -0.0625 The t term is 2.5t. Take half its coefficient (1.25). Square it (1.5625) and add it to both sides. Add '1.5625' to each side of the equation. 2.5t + 1.5625 + t2 = -0.0625 + 1.5625 Reorder the terms: 1.5625 + 2.5t + t2 = -0.0625 + 1.5625 Combine like terms: -0.0625 + 1.5625 = 1.5 1.5625 + 2.5t + t2 = 1.5 Factor a perfect square on the left side: (t + 1.25)(t + 1.25) = 1.5 Calculate the square root of the right side: 1.224744871 Break this problem into two subproblems by setting (t + 1.25) equal to 1.224744871 and -1.224744871.Subproblem 1
t + 1.25 = 1.224744871 Simplifying t + 1.25 = 1.224744871 Reorder the terms: 1.25 + t = 1.224744871 Solving 1.25 + t = 1.224744871 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + t = 1.224744871 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + t = 1.224744871 + -1.25 t = 1.224744871 + -1.25 Combine like terms: 1.224744871 + -1.25 = -0.025255129 t = -0.025255129 Simplifying t = -0.025255129Subproblem 2
t + 1.25 = -1.224744871 Simplifying t + 1.25 = -1.224744871 Reorder the terms: 1.25 + t = -1.224744871 Solving 1.25 + t = -1.224744871 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + t = -1.224744871 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + t = -1.224744871 + -1.25 t = -1.224744871 + -1.25 Combine like terms: -1.224744871 + -1.25 = -2.474744871 t = -2.474744871 Simplifying t = -2.474744871Solution
The solution to the problem is based on the solutions from the subproblems. t = {-0.025255129, -2.474744871}Solution
t = {-0.025255129, -2.474744871}
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