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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 = 0.375 0.5625 + -1.5t + t2 = 0.375 Factor a perfect square on the left side: (t + -0.75)(t + -0.75) = 0.375 Calculate the square root of the right side: 0.612372436 Break this problem into two subproblems by setting (t + -0.75) equal to 0.612372436 and -0.612372436.Subproblem 1
t + -0.75 = 0.612372436 Simplifying t + -0.75 = 0.612372436 Reorder the terms: -0.75 + t = 0.612372436 Solving -0.75 + t = 0.612372436 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 = 0.612372436 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + t = 0.612372436 + 0.75 t = 0.612372436 + 0.75 Combine like terms: 0.612372436 + 0.75 = 1.362372436 t = 1.362372436 Simplifying t = 1.362372436Subproblem 2
t + -0.75 = -0.612372436 Simplifying t + -0.75 = -0.612372436 Reorder the terms: -0.75 + t = -0.612372436 Solving -0.75 + t = -0.612372436 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 = -0.612372436 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + t = -0.612372436 + 0.75 t = -0.612372436 + 0.75 Combine like terms: -0.612372436 + 0.75 = 0.137627564 t = 0.137627564 Simplifying t = 0.137627564Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.362372436, 0.137627564}
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