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Simplifying 90 = -16t2 + 48t + 64 Reorder the terms: 90 = 64 + 48t + -16t2 Solving 90 = 64 + 48t + -16t2 Solving for variable 't'. Combine like terms: 90 + -64 = 26 26 + -48t + 16t2 = 64 + 48t + -16t2 + -64 + -48t + 16t2 Reorder the terms: 26 + -48t + 16t2 = 64 + -64 + 48t + -48t + -16t2 + 16t2 Combine like terms: 64 + -64 = 0 26 + -48t + 16t2 = 0 + 48t + -48t + -16t2 + 16t2 26 + -48t + 16t2 = 48t + -48t + -16t2 + 16t2 Combine like terms: 48t + -48t = 0 26 + -48t + 16t2 = 0 + -16t2 + 16t2 26 + -48t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 26 + -48t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(13 + -24t + 8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(13 + -24t + 8t2)' equal to zero and attempt to solve: Simplifying 13 + -24t + 8t2 = 0 Solving 13 + -24t + 8t2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. 1.625 + -3t + t2 = 0 Move the constant term to the right: Add '-1.625' to each side of the equation. 1.625 + -3t + -1.625 + t2 = 0 + -1.625 Reorder the terms: 1.625 + -1.625 + -3t + t2 = 0 + -1.625 Combine like terms: 1.625 + -1.625 = 0.000 0.000 + -3t + t2 = 0 + -1.625 -3t + t2 = 0 + -1.625 Combine like terms: 0 + -1.625 = -1.625 -3t + t2 = -1.625 The t term is -3t. Take half its coefficient (-1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. -3t + 2.25 + t2 = -1.625 + 2.25 Reorder the terms: 2.25 + -3t + t2 = -1.625 + 2.25 Combine like terms: -1.625 + 2.25 = 0.625 2.25 + -3t + t2 = 0.625 Factor a perfect square on the left side: (t + -1.5)(t + -1.5) = 0.625 Calculate the square root of the right side: 0.790569415 Break this problem into two subproblems by setting (t + -1.5) equal to 0.790569415 and -0.790569415.Subproblem 1
t + -1.5 = 0.790569415 Simplifying t + -1.5 = 0.790569415 Reorder the terms: -1.5 + t = 0.790569415 Solving -1.5 + t = 0.790569415 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + t = 0.790569415 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + t = 0.790569415 + 1.5 t = 0.790569415 + 1.5 Combine like terms: 0.790569415 + 1.5 = 2.290569415 t = 2.290569415 Simplifying t = 2.290569415Subproblem 2
t + -1.5 = -0.790569415 Simplifying t + -1.5 = -0.790569415 Reorder the terms: -1.5 + t = -0.790569415 Solving -1.5 + t = -0.790569415 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + t = -0.790569415 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + t = -0.790569415 + 1.5 t = -0.790569415 + 1.5 Combine like terms: -0.790569415 + 1.5 = 0.709430585 t = 0.709430585 Simplifying t = 0.709430585Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.290569415, 0.709430585}Solution
t = {2.290569415, 0.709430585}
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