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Simplifying 12 = -16t2 + 100t + 2 Reorder the terms: 12 = 2 + 100t + -16t2 Solving 12 = 2 + 100t + -16t2 Solving for variable 't'. Combine like terms: 12 + -2 = 10 10 + -100t + 16t2 = 2 + 100t + -16t2 + -2 + -100t + 16t2 Reorder the terms: 10 + -100t + 16t2 = 2 + -2 + 100t + -100t + -16t2 + 16t2 Combine like terms: 2 + -2 = 0 10 + -100t + 16t2 = 0 + 100t + -100t + -16t2 + 16t2 10 + -100t + 16t2 = 100t + -100t + -16t2 + 16t2 Combine like terms: 100t + -100t = 0 10 + -100t + 16t2 = 0 + -16t2 + 16t2 10 + -100t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 10 + -100t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(5 + -50t + 8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(5 + -50t + 8t2)' equal to zero and attempt to solve: Simplifying 5 + -50t + 8t2 = 0 Solving 5 + -50t + 8t2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. 0.625 + -6.25t + t2 = 0 Move the constant term to the right: Add '-0.625' to each side of the equation. 0.625 + -6.25t + -0.625 + t2 = 0 + -0.625 Reorder the terms: 0.625 + -0.625 + -6.25t + t2 = 0 + -0.625 Combine like terms: 0.625 + -0.625 = 0.000 0.000 + -6.25t + t2 = 0 + -0.625 -6.25t + t2 = 0 + -0.625 Combine like terms: 0 + -0.625 = -0.625 -6.25t + t2 = -0.625 The t term is -6.25t. Take half its coefficient (-3.125). Square it (9.765625) and add it to both sides. Add '9.765625' to each side of the equation. -6.25t + 9.765625 + t2 = -0.625 + 9.765625 Reorder the terms: 9.765625 + -6.25t + t2 = -0.625 + 9.765625 Combine like terms: -0.625 + 9.765625 = 9.140625 9.765625 + -6.25t + t2 = 9.140625 Factor a perfect square on the left side: (t + -3.125)(t + -3.125) = 9.140625 Calculate the square root of the right side: 3.023346656 Break this problem into two subproblems by setting (t + -3.125) equal to 3.023346656 and -3.023346656.Subproblem 1
t + -3.125 = 3.023346656 Simplifying t + -3.125 = 3.023346656 Reorder the terms: -3.125 + t = 3.023346656 Solving -3.125 + t = 3.023346656 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3.125' to each side of the equation. -3.125 + 3.125 + t = 3.023346656 + 3.125 Combine like terms: -3.125 + 3.125 = 0.000 0.000 + t = 3.023346656 + 3.125 t = 3.023346656 + 3.125 Combine like terms: 3.023346656 + 3.125 = 6.148346656 t = 6.148346656 Simplifying t = 6.148346656Subproblem 2
t + -3.125 = -3.023346656 Simplifying t + -3.125 = -3.023346656 Reorder the terms: -3.125 + t = -3.023346656 Solving -3.125 + t = -3.023346656 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3.125' to each side of the equation. -3.125 + 3.125 + t = -3.023346656 + 3.125 Combine like terms: -3.125 + 3.125 = 0.000 0.000 + t = -3.023346656 + 3.125 t = -3.023346656 + 3.125 Combine like terms: -3.023346656 + 3.125 = 0.101653344 t = 0.101653344 Simplifying t = 0.101653344Solution
The solution to the problem is based on the solutions from the subproblems. t = {6.148346656, 0.101653344}Solution
t = {6.148346656, 0.101653344}
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