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Simplifying 24 = 72t + -16t2 Solving 24 = 72t + -16t2 Solving for variable 't'. Reorder the terms: 24 + -72t + 16t2 = 72t + -72t + -16t2 + 16t2 Combine like terms: 72t + -72t = 0 24 + -72t + 16t2 = 0 + -16t2 + 16t2 24 + -72t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 24 + -72t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '8'. 8(3 + -9t + 2t2) = 0 Ignore the factor 8.Subproblem 1
Set the factor '(3 + -9t + 2t2)' equal to zero and attempt to solve: Simplifying 3 + -9t + 2t2 = 0 Solving 3 + -9t + 2t2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 1.5 + -4.5t + t2 = 0 Move the constant term to the right: Add '-1.5' to each side of the equation. 1.5 + -4.5t + -1.5 + t2 = 0 + -1.5 Reorder the terms: 1.5 + -1.5 + -4.5t + t2 = 0 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + -4.5t + t2 = 0 + -1.5 -4.5t + t2 = 0 + -1.5 Combine like terms: 0 + -1.5 = -1.5 -4.5t + t2 = -1.5 The t term is -4.5t. Take half its coefficient (-2.25). Square it (5.0625) and add it to both sides. Add '5.0625' to each side of the equation. -4.5t + 5.0625 + t2 = -1.5 + 5.0625 Reorder the terms: 5.0625 + -4.5t + t2 = -1.5 + 5.0625 Combine like terms: -1.5 + 5.0625 = 3.5625 5.0625 + -4.5t + t2 = 3.5625 Factor a perfect square on the left side: (t + -2.25)(t + -2.25) = 3.5625 Calculate the square root of the right side: 1.887458609 Break this problem into two subproblems by setting (t + -2.25) equal to 1.887458609 and -1.887458609.Subproblem 1
t + -2.25 = 1.887458609 Simplifying t + -2.25 = 1.887458609 Reorder the terms: -2.25 + t = 1.887458609 Solving -2.25 + t = 1.887458609 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.25' to each side of the equation. -2.25 + 2.25 + t = 1.887458609 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + t = 1.887458609 + 2.25 t = 1.887458609 + 2.25 Combine like terms: 1.887458609 + 2.25 = 4.137458609 t = 4.137458609 Simplifying t = 4.137458609Subproblem 2
t + -2.25 = -1.887458609 Simplifying t + -2.25 = -1.887458609 Reorder the terms: -2.25 + t = -1.887458609 Solving -2.25 + t = -1.887458609 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.25' to each side of the equation. -2.25 + 2.25 + t = -1.887458609 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + t = -1.887458609 + 2.25 t = -1.887458609 + 2.25 Combine like terms: -1.887458609 + 2.25 = 0.362541391 t = 0.362541391 Simplifying t = 0.362541391Solution
The solution to the problem is based on the solutions from the subproblems. t = {4.137458609, 0.362541391}Solution
t = {4.137458609, 0.362541391}
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