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Simplifying 13.2 = -16t2 + 24t + 16.4 Reorder the terms: 13.2 = 16.4 + 24t + -16t2 Solving 13.2 = 16.4 + 24t + -16t2 Solving for variable 't'. Combine like terms: 13.2 + -16.4 = -3.2 -3.2 + -24t + 16t2 = 16.4 + 24t + -16t2 + -16.4 + -24t + 16t2 Reorder the terms: -3.2 + -24t + 16t2 = 16.4 + -16.4 + 24t + -24t + -16t2 + 16t2 Combine like terms: 16.4 + -16.4 = 0.0 -3.2 + -24t + 16t2 = 0.0 + 24t + -24t + -16t2 + 16t2 -3.2 + -24t + 16t2 = 24t + -24t + -16t2 + 16t2 Combine like terms: 24t + -24t = 0 -3.2 + -24t + 16t2 = 0 + -16t2 + 16t2 -3.2 + -24t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -3.2 + -24t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -0.2 + -1.5t + t2 = 0 Move the constant term to the right: Add '0.2' to each side of the equation. -0.2 + -1.5t + 0.2 + t2 = 0 + 0.2 Reorder the terms: -0.2 + 0.2 + -1.5t + t2 = 0 + 0.2 Combine like terms: -0.2 + 0.2 = 0.0 0.0 + -1.5t + t2 = 0 + 0.2 -1.5t + t2 = 0 + 0.2 Combine like terms: 0 + 0.2 = 0.2 -1.5t + t2 = 0.2 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 = 0.2 + 0.5625 Reorder the terms: 0.5625 + -1.5t + t2 = 0.2 + 0.5625 Combine like terms: 0.2 + 0.5625 = 0.7625 0.5625 + -1.5t + t2 = 0.7625 Factor a perfect square on the left side: (t + -0.75)(t + -0.75) = 0.7625 Calculate the square root of the right side: 0.87321246 Break this problem into two subproblems by setting (t + -0.75) equal to 0.87321246 and -0.87321246.Subproblem 1
t + -0.75 = 0.87321246 Simplifying t + -0.75 = 0.87321246 Reorder the terms: -0.75 + t = 0.87321246 Solving -0.75 + t = 0.87321246 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 = 0.87321246 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + t = 0.87321246 + 0.75 t = 0.87321246 + 0.75 Combine like terms: 0.87321246 + 0.75 = 1.62321246 t = 1.62321246 Simplifying t = 1.62321246Subproblem 2
t + -0.75 = -0.87321246 Simplifying t + -0.75 = -0.87321246 Reorder the terms: -0.75 + t = -0.87321246 Solving -0.75 + t = -0.87321246 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 = -0.87321246 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + t = -0.87321246 + 0.75 t = -0.87321246 + 0.75 Combine like terms: -0.87321246 + 0.75 = -0.12321246 t = -0.12321246 Simplifying t = -0.12321246Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.62321246, -0.12321246}
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