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Simplifying 0 = -16t2 + 24t + 7.5 Reorder the terms: 0 = 7.5 + 24t + -16t2 Solving 0 = 7.5 + 24t + -16t2 Solving for variable 't'. Combine like terms: 0 + -7.5 = -7.5 -7.5 + -24t + 16t2 = 7.5 + 24t + -16t2 + -7.5 + -24t + 16t2 Reorder the terms: -7.5 + -24t + 16t2 = 7.5 + -7.5 + 24t + -24t + -16t2 + 16t2 Combine like terms: 7.5 + -7.5 = 0.0 -7.5 + -24t + 16t2 = 0.0 + 24t + -24t + -16t2 + 16t2 -7.5 + -24t + 16t2 = 24t + -24t + -16t2 + 16t2 Combine like terms: 24t + -24t = 0 -7.5 + -24t + 16t2 = 0 + -16t2 + 16t2 -7.5 + -24t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -7.5 + -24t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -0.46875 + -1.5t + t2 = 0 Move the constant term to the right: Add '0.46875' to each side of the equation. -0.46875 + -1.5t + 0.46875 + t2 = 0 + 0.46875 Reorder the terms: -0.46875 + 0.46875 + -1.5t + t2 = 0 + 0.46875 Combine like terms: -0.46875 + 0.46875 = 0.00000 0.00000 + -1.5t + t2 = 0 + 0.46875 -1.5t + t2 = 0 + 0.46875 Combine like terms: 0 + 0.46875 = 0.46875 -1.5t + t2 = 0.46875 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 = 0.46875 + 0.5625 Reorder the terms: 0.5625 + -1.5t + t2 = 0.46875 + 0.5625 Combine like terms: 0.46875 + 0.5625 = 1.03125 0.5625 + -1.5t + t2 = 1.03125 Factor a perfect square on the left side: (t + -0.75)(t + -0.75) = 1.03125 Calculate the square root of the right side: 1.015504801 Break this problem into two subproblems by setting (t + -0.75) equal to 1.015504801 and -1.015504801.Subproblem 1
t + -0.75 = 1.015504801 Simplifying t + -0.75 = 1.015504801 Reorder the terms: -0.75 + t = 1.015504801 Solving -0.75 + t = 1.015504801 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 = 1.015504801 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + t = 1.015504801 + 0.75 t = 1.015504801 + 0.75 Combine like terms: 1.015504801 + 0.75 = 1.765504801 t = 1.765504801 Simplifying t = 1.765504801Subproblem 2
t + -0.75 = -1.015504801 Simplifying t + -0.75 = -1.015504801 Reorder the terms: -0.75 + t = -1.015504801 Solving -0.75 + t = -1.015504801 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 = -1.015504801 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + t = -1.015504801 + 0.75 t = -1.015504801 + 0.75 Combine like terms: -1.015504801 + 0.75 = -0.265504801 t = -0.265504801 Simplifying t = -0.265504801Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.765504801, -0.265504801}
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