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Simplifying 23.76 = -16t2 + 12t + 22 Reorder the terms: 23.76 = 22 + 12t + -16t2 Solving 23.76 = 22 + 12t + -16t2 Solving for variable 't'. Combine like terms: 23.76 + -22 = 1.76 1.76 + -12t + 16t2 = 22 + 12t + -16t2 + -22 + -12t + 16t2 Reorder the terms: 1.76 + -12t + 16t2 = 22 + -22 + 12t + -12t + -16t2 + 16t2 Combine like terms: 22 + -22 = 0 1.76 + -12t + 16t2 = 0 + 12t + -12t + -16t2 + 16t2 1.76 + -12t + 16t2 = 12t + -12t + -16t2 + 16t2 Combine like terms: 12t + -12t = 0 1.76 + -12t + 16t2 = 0 + -16t2 + 16t2 1.76 + -12t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 1.76 + -12t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 0.11 + -0.75t + t2 = 0 Move the constant term to the right: Add '-0.11' to each side of the equation. 0.11 + -0.75t + -0.11 + t2 = 0 + -0.11 Reorder the terms: 0.11 + -0.11 + -0.75t + t2 = 0 + -0.11 Combine like terms: 0.11 + -0.11 = 0.00 0.00 + -0.75t + t2 = 0 + -0.11 -0.75t + t2 = 0 + -0.11 Combine like terms: 0 + -0.11 = -0.11 -0.75t + t2 = -0.11 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 = -0.11 + 0.140625 Reorder the terms: 0.140625 + -0.75t + t2 = -0.11 + 0.140625 Combine like terms: -0.11 + 0.140625 = 0.030625 0.140625 + -0.75t + t2 = 0.030625 Factor a perfect square on the left side: (t + -0.375)(t + -0.375) = 0.030625 Calculate the square root of the right side: 0.175 Break this problem into two subproblems by setting (t + -0.375) equal to 0.175 and -0.175.Subproblem 1
t + -0.375 = 0.175 Simplifying t + -0.375 = 0.175 Reorder the terms: -0.375 + t = 0.175 Solving -0.375 + t = 0.175 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 = 0.175 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + t = 0.175 + 0.375 t = 0.175 + 0.375 Combine like terms: 0.175 + 0.375 = 0.55 t = 0.55 Simplifying t = 0.55Subproblem 2
t + -0.375 = -0.175 Simplifying t + -0.375 = -0.175 Reorder the terms: -0.375 + t = -0.175 Solving -0.375 + t = -0.175 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 = -0.175 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + t = -0.175 + 0.375 t = -0.175 + 0.375 Combine like terms: -0.175 + 0.375 = 0.2 t = 0.2 Simplifying t = 0.2Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.55, 0.2}
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