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Simplifying 24.25 = -16t2 + 12t + 22 Reorder the terms: 24.25 = 22 + 12t + -16t2 Solving 24.25 = 22 + 12t + -16t2 Solving for variable 't'. Combine like terms: 24.25 + -22 = 2.25 2.25 + -12t + 16t2 = 22 + 12t + -16t2 + -22 + -12t + 16t2 Reorder the terms: 2.25 + -12t + 16t2 = 22 + -22 + 12t + -12t + -16t2 + 16t2 Combine like terms: 22 + -22 = 0 2.25 + -12t + 16t2 = 0 + 12t + -12t + -16t2 + 16t2 2.25 + -12t + 16t2 = 12t + -12t + -16t2 + 16t2 Combine like terms: 12t + -12t = 0 2.25 + -12t + 16t2 = 0 + -16t2 + 16t2 2.25 + -12t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 2.25 + -12t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 0.140625 + -0.75t + t2 = 0 Move the constant term to the right: Add '-0.140625' to each side of the equation. 0.140625 + -0.75t + -0.140625 + t2 = 0 + -0.140625 Reorder the terms: 0.140625 + -0.140625 + -0.75t + t2 = 0 + -0.140625 Combine like terms: 0.140625 + -0.140625 = 0.000000 0.000000 + -0.75t + t2 = 0 + -0.140625 -0.75t + t2 = 0 + -0.140625 Combine like terms: 0 + -0.140625 = -0.140625 -0.75t + t2 = -0.140625 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 = -0.140625 + 0.140625 Reorder the terms: 0.140625 + -0.75t + t2 = -0.140625 + 0.140625 Combine like terms: -0.140625 + 0.140625 = 0.000000 0.140625 + -0.75t + t2 = 0.000000 Factor a perfect square on the left side: (t + -0.375)(t + -0.375) = 0.000000 Calculate the square root of the right side: 0.000000 Break this problem into two subproblems by setting (t + -0.375) equal to 0.000000 and 0.000000.Subproblem 1
t + -0.375 = 0.000000 Simplifying t + -0.375 = 0.000000 Reorder the terms: -0.375 + t = 0.000000 Solving -0.375 + t = 0.000000 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 = 0.000000 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + t = 0.000000 + 0.375 t = 0.000000 + 0.375 Combine like terms: 0.000000 + 0.375 = 0.375 t = 0.375 Simplifying t = 0.375Subproblem 2
t + -0.375 = 0.000000 Simplifying t + -0.375 = 0.000000 Reorder the terms: -0.375 + t = 0.000000 Solving -0.375 + t = 0.000000 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 = 0.000000 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + t = 0.000000 + 0.375 t = 0.000000 + 0.375 Combine like terms: 0.000000 + 0.375 = 0.375 t = 0.375 Simplifying t = 0.375Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.375, 0.375}
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