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Simplifying 0 = -16t2 + 32t + 3000 Reorder the terms: 0 = 3000 + 32t + -16t2 Solving 0 = 3000 + 32t + -16t2 Solving for variable 't'. Combine like terms: 0 + -3000 = -3000 -3000 + -32t + 16t2 = 3000 + 32t + -16t2 + -3000 + -32t + 16t2 Reorder the terms: -3000 + -32t + 16t2 = 3000 + -3000 + 32t + -32t + -16t2 + 16t2 Combine like terms: 3000 + -3000 = 0 -3000 + -32t + 16t2 = 0 + 32t + -32t + -16t2 + 16t2 -3000 + -32t + 16t2 = 32t + -32t + -16t2 + 16t2 Combine like terms: 32t + -32t = 0 -3000 + -32t + 16t2 = 0 + -16t2 + 16t2 -3000 + -32t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -3000 + -32t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '8'. 8(-375 + -4t + 2t2) = 0 Ignore the factor 8.Subproblem 1
Set the factor '(-375 + -4t + 2t2)' equal to zero and attempt to solve: Simplifying -375 + -4t + 2t2 = 0 Solving -375 + -4t + 2t2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -187.5 + -2t + t2 = 0 Move the constant term to the right: Add '187.5' to each side of the equation. -187.5 + -2t + 187.5 + t2 = 0 + 187.5 Reorder the terms: -187.5 + 187.5 + -2t + t2 = 0 + 187.5 Combine like terms: -187.5 + 187.5 = 0.0 0.0 + -2t + t2 = 0 + 187.5 -2t + t2 = 0 + 187.5 Combine like terms: 0 + 187.5 = 187.5 -2t + t2 = 187.5 The t term is -2t. Take half its coefficient (-1). Square it (1) and add it to both sides. Add '1' to each side of the equation. -2t + 1 + t2 = 187.5 + 1 Reorder the terms: 1 + -2t + t2 = 187.5 + 1 Combine like terms: 187.5 + 1 = 188.5 1 + -2t + t2 = 188.5 Factor a perfect square on the left side: (t + -1)(t + -1) = 188.5 Calculate the square root of the right side: 13.729530218 Break this problem into two subproblems by setting (t + -1) equal to 13.729530218 and -13.729530218.Subproblem 1
t + -1 = 13.729530218 Simplifying t + -1 = 13.729530218 Reorder the terms: -1 + t = 13.729530218 Solving -1 + t = 13.729530218 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + t = 13.729530218 + 1 Combine like terms: -1 + 1 = 0 0 + t = 13.729530218 + 1 t = 13.729530218 + 1 Combine like terms: 13.729530218 + 1 = 14.729530218 t = 14.729530218 Simplifying t = 14.729530218Subproblem 2
t + -1 = -13.729530218 Simplifying t + -1 = -13.729530218 Reorder the terms: -1 + t = -13.729530218 Solving -1 + t = -13.729530218 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + t = -13.729530218 + 1 Combine like terms: -1 + 1 = 0 0 + t = -13.729530218 + 1 t = -13.729530218 + 1 Combine like terms: -13.729530218 + 1 = -12.729530218 t = -12.729530218 Simplifying t = -12.729530218Solution
The solution to the problem is based on the solutions from the subproblems. t = {14.729530218, -12.729530218}Solution
t = {14.729530218, -12.729530218}
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