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Simplifying 0 = -16t2 + 128t + -198 Reorder the terms: 0 = -198 + 128t + -16t2 Solving 0 = -198 + 128t + -16t2 Solving for variable 't'. Combine like terms: 0 + 198 = 198 198 + -128t + 16t2 = -198 + 128t + -16t2 + 198 + -128t + 16t2 Reorder the terms: 198 + -128t + 16t2 = -198 + 198 + 128t + -128t + -16t2 + 16t2 Combine like terms: -198 + 198 = 0 198 + -128t + 16t2 = 0 + 128t + -128t + -16t2 + 16t2 198 + -128t + 16t2 = 128t + -128t + -16t2 + 16t2 Combine like terms: 128t + -128t = 0 198 + -128t + 16t2 = 0 + -16t2 + 16t2 198 + -128t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 198 + -128t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(99 + -64t + 8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(99 + -64t + 8t2)' equal to zero and attempt to solve: Simplifying 99 + -64t + 8t2 = 0 Solving 99 + -64t + 8t2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. 12.375 + -8t + t2 = 0 Move the constant term to the right: Add '-12.375' to each side of the equation. 12.375 + -8t + -12.375 + t2 = 0 + -12.375 Reorder the terms: 12.375 + -12.375 + -8t + t2 = 0 + -12.375 Combine like terms: 12.375 + -12.375 = 0.000 0.000 + -8t + t2 = 0 + -12.375 -8t + t2 = 0 + -12.375 Combine like terms: 0 + -12.375 = -12.375 -8t + t2 = -12.375 The t term is -8t. Take half its coefficient (-4). Square it (16) and add it to both sides. Add '16' to each side of the equation. -8t + 16 + t2 = -12.375 + 16 Reorder the terms: 16 + -8t + t2 = -12.375 + 16 Combine like terms: -12.375 + 16 = 3.625 16 + -8t + t2 = 3.625 Factor a perfect square on the left side: (t + -4)(t + -4) = 3.625 Calculate the square root of the right side: 1.903943276 Break this problem into two subproblems by setting (t + -4) equal to 1.903943276 and -1.903943276.Subproblem 1
t + -4 = 1.903943276 Simplifying t + -4 = 1.903943276 Reorder the terms: -4 + t = 1.903943276 Solving -4 + t = 1.903943276 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + t = 1.903943276 + 4 Combine like terms: -4 + 4 = 0 0 + t = 1.903943276 + 4 t = 1.903943276 + 4 Combine like terms: 1.903943276 + 4 = 5.903943276 t = 5.903943276 Simplifying t = 5.903943276Subproblem 2
t + -4 = -1.903943276 Simplifying t + -4 = -1.903943276 Reorder the terms: -4 + t = -1.903943276 Solving -4 + t = -1.903943276 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + t = -1.903943276 + 4 Combine like terms: -4 + 4 = 0 0 + t = -1.903943276 + 4 t = -1.903943276 + 4 Combine like terms: -1.903943276 + 4 = 2.096056724 t = 2.096056724 Simplifying t = 2.096056724Solution
The solution to the problem is based on the solutions from the subproblems. t = {5.903943276, 2.096056724}Solution
t = {5.903943276, 2.096056724}
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