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Simplifying 96t + -16t2 = 40 Solving 96t + -16t2 = 40 Solving for variable 't'. Reorder the terms: -40 + 96t + -16t2 = 40 + -40 Combine like terms: 40 + -40 = 0 -40 + 96t + -16t2 = 0 Factor out the Greatest Common Factor (GCF), '8'. 8(-5 + 12t + -2t2) = 0 Ignore the factor 8.Subproblem 1
Set the factor '(-5 + 12t + -2t2)' equal to zero and attempt to solve: Simplifying -5 + 12t + -2t2 = 0 Solving -5 + 12t + -2t2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. 2.5 + -6t + t2 = 0 Move the constant term to the right: Add '-2.5' to each side of the equation. 2.5 + -6t + -2.5 + t2 = 0 + -2.5 Reorder the terms: 2.5 + -2.5 + -6t + t2 = 0 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + -6t + t2 = 0 + -2.5 -6t + t2 = 0 + -2.5 Combine like terms: 0 + -2.5 = -2.5 -6t + t2 = -2.5 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 = -2.5 + 9 Reorder the terms: 9 + -6t + t2 = -2.5 + 9 Combine like terms: -2.5 + 9 = 6.5 9 + -6t + t2 = 6.5 Factor a perfect square on the left side: (t + -3)(t + -3) = 6.5 Calculate the square root of the right side: 2.549509757 Break this problem into two subproblems by setting (t + -3) equal to 2.549509757 and -2.549509757.Subproblem 1
t + -3 = 2.549509757 Simplifying t + -3 = 2.549509757 Reorder the terms: -3 + t = 2.549509757 Solving -3 + t = 2.549509757 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 = 2.549509757 + 3 Combine like terms: -3 + 3 = 0 0 + t = 2.549509757 + 3 t = 2.549509757 + 3 Combine like terms: 2.549509757 + 3 = 5.549509757 t = 5.549509757 Simplifying t = 5.549509757Subproblem 2
t + -3 = -2.549509757 Simplifying t + -3 = -2.549509757 Reorder the terms: -3 + t = -2.549509757 Solving -3 + t = -2.549509757 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 = -2.549509757 + 3 Combine like terms: -3 + 3 = 0 0 + t = -2.549509757 + 3 t = -2.549509757 + 3 Combine like terms: -2.549509757 + 3 = 0.450490243 t = 0.450490243 Simplifying t = 0.450490243Solution
The solution to the problem is based on the solutions from the subproblems. t = {5.549509757, 0.450490243}Solution
t = {5.549509757, 0.450490243}
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