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Simplifying 0 = 160 + -12t + -16t2 Solving 0 = 160 + -12t + -16t2 Solving for variable 't'. Combine like terms: 0 + -160 = -160 -160 + 12t + 16t2 = 160 + -12t + -16t2 + -160 + 12t + 16t2 Reorder the terms: -160 + 12t + 16t2 = 160 + -160 + -12t + 12t + -16t2 + 16t2 Combine like terms: 160 + -160 = 0 -160 + 12t + 16t2 = 0 + -12t + 12t + -16t2 + 16t2 -160 + 12t + 16t2 = -12t + 12t + -16t2 + 16t2 Combine like terms: -12t + 12t = 0 -160 + 12t + 16t2 = 0 + -16t2 + 16t2 -160 + 12t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -160 + 12t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(-40 + 3t + 4t2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(-40 + 3t + 4t2)' equal to zero and attempt to solve: Simplifying -40 + 3t + 4t2 = 0 Solving -40 + 3t + 4t2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -10 + 0.75t + t2 = 0 Move the constant term to the right: Add '10' to each side of the equation. -10 + 0.75t + 10 + t2 = 0 + 10 Reorder the terms: -10 + 10 + 0.75t + t2 = 0 + 10 Combine like terms: -10 + 10 = 0 0 + 0.75t + t2 = 0 + 10 0.75t + t2 = 0 + 10 Combine like terms: 0 + 10 = 10 0.75t + t2 = 10 The t term is 0.75t. Take half its coefficient (0.375). Square it (0.140625) and add it to both sides. Add '0.140625' to each side of the equation. 0.75t + 0.140625 + t2 = 10 + 0.140625 Reorder the terms: 0.140625 + 0.75t + t2 = 10 + 0.140625 Combine like terms: 10 + 0.140625 = 10.140625 0.140625 + 0.75t + t2 = 10.140625 Factor a perfect square on the left side: (t + 0.375)(t + 0.375) = 10.140625 Calculate the square root of the right side: 3.184434801 Break this problem into two subproblems by setting (t + 0.375) equal to 3.184434801 and -3.184434801.Subproblem 1
t + 0.375 = 3.184434801 Simplifying t + 0.375 = 3.184434801 Reorder the terms: 0.375 + t = 3.184434801 Solving 0.375 + t = 3.184434801 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.375' to each side of the equation. 0.375 + -0.375 + t = 3.184434801 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + t = 3.184434801 + -0.375 t = 3.184434801 + -0.375 Combine like terms: 3.184434801 + -0.375 = 2.809434801 t = 2.809434801 Simplifying t = 2.809434801Subproblem 2
t + 0.375 = -3.184434801 Simplifying t + 0.375 = -3.184434801 Reorder the terms: 0.375 + t = -3.184434801 Solving 0.375 + t = -3.184434801 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.375' to each side of the equation. 0.375 + -0.375 + t = -3.184434801 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + t = -3.184434801 + -0.375 t = -3.184434801 + -0.375 Combine like terms: -3.184434801 + -0.375 = -3.559434801 t = -3.559434801 Simplifying t = -3.559434801Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.809434801, -3.559434801}Solution
t = {2.809434801, -3.559434801}
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