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Simplifying 0 = -16t2 + 24t + 3 Reorder the terms: 0 = 3 + 24t + -16t2 Solving 0 = 3 + 24t + -16t2 Solving for variable 't'. Combine like terms: 0 + -3 = -3 -3 + -24t + 16t2 = 3 + 24t + -16t2 + -3 + -24t + 16t2 Reorder the terms: -3 + -24t + 16t2 = 3 + -3 + 24t + -24t + -16t2 + 16t2 Combine like terms: 3 + -3 = 0 -3 + -24t + 16t2 = 0 + 24t + -24t + -16t2 + 16t2 -3 + -24t + 16t2 = 24t + -24t + -16t2 + 16t2 Combine like terms: 24t + -24t = 0 -3 + -24t + 16t2 = 0 + -16t2 + 16t2 -3 + -24t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -3 + -24t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -0.1875 + -1.5t + t2 = 0 Move the constant term to the right: Add '0.1875' to each side of the equation. -0.1875 + -1.5t + 0.1875 + t2 = 0 + 0.1875 Reorder the terms: -0.1875 + 0.1875 + -1.5t + t2 = 0 + 0.1875 Combine like terms: -0.1875 + 0.1875 = 0.0000 0.0000 + -1.5t + t2 = 0 + 0.1875 -1.5t + t2 = 0 + 0.1875 Combine like terms: 0 + 0.1875 = 0.1875 -1.5t + t2 = 0.1875 The t term is -1.5t. Take half its coefficient (-0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. -1.5t + 0.5625 + t2 = 0.1875 + 0.5625 Reorder the terms: 0.5625 + -1.5t + t2 = 0.1875 + 0.5625 Combine like terms: 0.1875 + 0.5625 = 0.75 0.5625 + -1.5t + t2 = 0.75 Factor a perfect square on the left side: (t + -0.75)(t + -0.75) = 0.75 Calculate the square root of the right side: 0.866025404 Break this problem into two subproblems by setting (t + -0.75) equal to 0.866025404 and -0.866025404.Subproblem 1
t + -0.75 = 0.866025404 Simplifying t + -0.75 = 0.866025404 Reorder the terms: -0.75 + t = 0.866025404 Solving -0.75 + t = 0.866025404 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + t = 0.866025404 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + t = 0.866025404 + 0.75 t = 0.866025404 + 0.75 Combine like terms: 0.866025404 + 0.75 = 1.616025404 t = 1.616025404 Simplifying t = 1.616025404Subproblem 2
t + -0.75 = -0.866025404 Simplifying t + -0.75 = -0.866025404 Reorder the terms: -0.75 + t = -0.866025404 Solving -0.75 + t = -0.866025404 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + t = -0.866025404 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + t = -0.866025404 + 0.75 t = -0.866025404 + 0.75 Combine like terms: -0.866025404 + 0.75 = -0.116025404 t = -0.116025404 Simplifying t = -0.116025404Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.616025404, -0.116025404}
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