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Simplifying 12 = -16t2 + 96t + 1152 Reorder the terms: 12 = 1152 + 96t + -16t2 Solving 12 = 1152 + 96t + -16t2 Solving for variable 't'. Combine like terms: 12 + -1152 = -1140 -1140 + -96t + 16t2 = 1152 + 96t + -16t2 + -1152 + -96t + 16t2 Reorder the terms: -1140 + -96t + 16t2 = 1152 + -1152 + 96t + -96t + -16t2 + 16t2 Combine like terms: 1152 + -1152 = 0 -1140 + -96t + 16t2 = 0 + 96t + -96t + -16t2 + 16t2 -1140 + -96t + 16t2 = 96t + -96t + -16t2 + 16t2 Combine like terms: 96t + -96t = 0 -1140 + -96t + 16t2 = 0 + -16t2 + 16t2 -1140 + -96t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -1140 + -96t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(-285 + -24t + 4t2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(-285 + -24t + 4t2)' equal to zero and attempt to solve: Simplifying -285 + -24t + 4t2 = 0 Solving -285 + -24t + 4t2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -71.25 + -6t + t2 = 0 Move the constant term to the right: Add '71.25' to each side of the equation. -71.25 + -6t + 71.25 + t2 = 0 + 71.25 Reorder the terms: -71.25 + 71.25 + -6t + t2 = 0 + 71.25 Combine like terms: -71.25 + 71.25 = 0.00 0.00 + -6t + t2 = 0 + 71.25 -6t + t2 = 0 + 71.25 Combine like terms: 0 + 71.25 = 71.25 -6t + t2 = 71.25 The t term is -6t. Take half its coefficient (-3). Square it (9) and add it to both sides. Add '9' to each side of the equation. -6t + 9 + t2 = 71.25 + 9 Reorder the terms: 9 + -6t + t2 = 71.25 + 9 Combine like terms: 71.25 + 9 = 80.25 9 + -6t + t2 = 80.25 Factor a perfect square on the left side: (t + -3)(t + -3) = 80.25 Calculate the square root of the right side: 8.958236434 Break this problem into two subproblems by setting (t + -3) equal to 8.958236434 and -8.958236434.Subproblem 1
t + -3 = 8.958236434 Simplifying t + -3 = 8.958236434 Reorder the terms: -3 + t = 8.958236434 Solving -3 + t = 8.958236434 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + t = 8.958236434 + 3 Combine like terms: -3 + 3 = 0 0 + t = 8.958236434 + 3 t = 8.958236434 + 3 Combine like terms: 8.958236434 + 3 = 11.958236434 t = 11.958236434 Simplifying t = 11.958236434Subproblem 2
t + -3 = -8.958236434 Simplifying t + -3 = -8.958236434 Reorder the terms: -3 + t = -8.958236434 Solving -3 + t = -8.958236434 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + t = -8.958236434 + 3 Combine like terms: -3 + 3 = 0 0 + t = -8.958236434 + 3 t = -8.958236434 + 3 Combine like terms: -8.958236434 + 3 = -5.958236434 t = -5.958236434 Simplifying t = -5.958236434Solution
The solution to the problem is based on the solutions from the subproblems. t = {11.958236434, -5.958236434}Solution
t = {11.958236434, -5.958236434}
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