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Simplifying 12 = -8t2 + 24t + 5 Reorder the terms: 12 = 5 + 24t + -8t2 Solving 12 = 5 + 24t + -8t2 Solving for variable 't'. Combine like terms: 12 + -5 = 7 7 + -24t + 8t2 = 5 + 24t + -8t2 + -5 + -24t + 8t2 Reorder the terms: 7 + -24t + 8t2 = 5 + -5 + 24t + -24t + -8t2 + 8t2 Combine like terms: 5 + -5 = 0 7 + -24t + 8t2 = 0 + 24t + -24t + -8t2 + 8t2 7 + -24t + 8t2 = 24t + -24t + -8t2 + 8t2 Combine like terms: 24t + -24t = 0 7 + -24t + 8t2 = 0 + -8t2 + 8t2 7 + -24t + 8t2 = -8t2 + 8t2 Combine like terms: -8t2 + 8t2 = 0 7 + -24t + 8t2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. 0.875 + -3t + t2 = 0 Move the constant term to the right: Add '-0.875' to each side of the equation. 0.875 + -3t + -0.875 + t2 = 0 + -0.875 Reorder the terms: 0.875 + -0.875 + -3t + t2 = 0 + -0.875 Combine like terms: 0.875 + -0.875 = 0.000 0.000 + -3t + t2 = 0 + -0.875 -3t + t2 = 0 + -0.875 Combine like terms: 0 + -0.875 = -0.875 -3t + t2 = -0.875 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 = -0.875 + 2.25 Reorder the terms: 2.25 + -3t + t2 = -0.875 + 2.25 Combine like terms: -0.875 + 2.25 = 1.375 2.25 + -3t + t2 = 1.375 Factor a perfect square on the left side: (t + -1.5)(t + -1.5) = 1.375 Calculate the square root of the right side: 1.17260394 Break this problem into two subproblems by setting (t + -1.5) equal to 1.17260394 and -1.17260394.Subproblem 1
t + -1.5 = 1.17260394 Simplifying t + -1.5 = 1.17260394 Reorder the terms: -1.5 + t = 1.17260394 Solving -1.5 + t = 1.17260394 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 = 1.17260394 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + t = 1.17260394 + 1.5 t = 1.17260394 + 1.5 Combine like terms: 1.17260394 + 1.5 = 2.67260394 t = 2.67260394 Simplifying t = 2.67260394Subproblem 2
t + -1.5 = -1.17260394 Simplifying t + -1.5 = -1.17260394 Reorder the terms: -1.5 + t = -1.17260394 Solving -1.5 + t = -1.17260394 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 = -1.17260394 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + t = -1.17260394 + 1.5 t = -1.17260394 + 1.5 Combine like terms: -1.17260394 + 1.5 = 0.32739606 t = 0.32739606 Simplifying t = 0.32739606Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.67260394, 0.32739606}
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