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Simplifying 0 = -16t2 + 12t + 1224 Reorder the terms: 0 = 1224 + 12t + -16t2 Solving 0 = 1224 + 12t + -16t2 Solving for variable 't'. Combine like terms: 0 + -1224 = -1224 -1224 + -12t + 16t2 = 1224 + 12t + -16t2 + -1224 + -12t + 16t2 Reorder the terms: -1224 + -12t + 16t2 = 1224 + -1224 + 12t + -12t + -16t2 + 16t2 Combine like terms: 1224 + -1224 = 0 -1224 + -12t + 16t2 = 0 + 12t + -12t + -16t2 + 16t2 -1224 + -12t + 16t2 = 12t + -12t + -16t2 + 16t2 Combine like terms: 12t + -12t = 0 -1224 + -12t + 16t2 = 0 + -16t2 + 16t2 -1224 + -12t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -1224 + -12t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(-306 + -3t + 4t2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(-306 + -3t + 4t2)' equal to zero and attempt to solve: Simplifying -306 + -3t + 4t2 = 0 Solving -306 + -3t + 4t2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -76.5 + -0.75t + t2 = 0 Move the constant term to the right: Add '76.5' to each side of the equation. -76.5 + -0.75t + 76.5 + t2 = 0 + 76.5 Reorder the terms: -76.5 + 76.5 + -0.75t + t2 = 0 + 76.5 Combine like terms: -76.5 + 76.5 = 0.0 0.0 + -0.75t + t2 = 0 + 76.5 -0.75t + t2 = 0 + 76.5 Combine like terms: 0 + 76.5 = 76.5 -0.75t + t2 = 76.5 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 = 76.5 + 0.140625 Reorder the terms: 0.140625 + -0.75t + t2 = 76.5 + 0.140625 Combine like terms: 76.5 + 0.140625 = 76.640625 0.140625 + -0.75t + t2 = 76.640625 Factor a perfect square on the left side: (t + -0.375)(t + -0.375) = 76.640625 Calculate the square root of the right side: 8.754463147 Break this problem into two subproblems by setting (t + -0.375) equal to 8.754463147 and -8.754463147.Subproblem 1
t + -0.375 = 8.754463147 Simplifying t + -0.375 = 8.754463147 Reorder the terms: -0.375 + t = 8.754463147 Solving -0.375 + t = 8.754463147 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 = 8.754463147 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + t = 8.754463147 + 0.375 t = 8.754463147 + 0.375 Combine like terms: 8.754463147 + 0.375 = 9.129463147 t = 9.129463147 Simplifying t = 9.129463147Subproblem 2
t + -0.375 = -8.754463147 Simplifying t + -0.375 = -8.754463147 Reorder the terms: -0.375 + t = -8.754463147 Solving -0.375 + t = -8.754463147 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 = -8.754463147 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + t = -8.754463147 + 0.375 t = -8.754463147 + 0.375 Combine like terms: -8.754463147 + 0.375 = -8.379463147 t = -8.379463147 Simplifying t = -8.379463147Solution
The solution to the problem is based on the solutions from the subproblems. t = {9.129463147, -8.379463147}Solution
t = {9.129463147, -8.379463147}
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