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Simplifying -16t2 + 12t + 25 = 0 Reorder the terms: 25 + 12t + -16t2 = 0 Solving 25 + 12t + -16t2 = 0 Solving for variable 't'. Begin completing the square. Divide all terms by -16 the coefficient of the squared term: Divide each side by '-16'. -1.5625 + -0.75t + t2 = 0 Move the constant term to the right: Add '1.5625' to each side of the equation. -1.5625 + -0.75t + 1.5625 + t2 = 0 + 1.5625 Reorder the terms: -1.5625 + 1.5625 + -0.75t + t2 = 0 + 1.5625 Combine like terms: -1.5625 + 1.5625 = 0.0000 0.0000 + -0.75t + t2 = 0 + 1.5625 -0.75t + t2 = 0 + 1.5625 Combine like terms: 0 + 1.5625 = 1.5625 -0.75t + t2 = 1.5625 The t term is -0.75t. Take half its coefficient (-0.375). Square it (0.140625) and add it to both sides. Add '0.140625' to each side of the equation. -0.75t + 0.140625 + t2 = 1.5625 + 0.140625 Reorder the terms: 0.140625 + -0.75t + t2 = 1.5625 + 0.140625 Combine like terms: 1.5625 + 0.140625 = 1.703125 0.140625 + -0.75t + t2 = 1.703125 Factor a perfect square on the left side: (t + -0.375)(t + -0.375) = 1.703125 Calculate the square root of the right side: 1.305038314 Break this problem into two subproblems by setting (t + -0.375) equal to 1.305038314 and -1.305038314.Subproblem 1
t + -0.375 = 1.305038314 Simplifying t + -0.375 = 1.305038314 Reorder the terms: -0.375 + t = 1.305038314 Solving -0.375 + t = 1.305038314 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.375' to each side of the equation. -0.375 + 0.375 + t = 1.305038314 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + t = 1.305038314 + 0.375 t = 1.305038314 + 0.375 Combine like terms: 1.305038314 + 0.375 = 1.680038314 t = 1.680038314 Simplifying t = 1.680038314Subproblem 2
t + -0.375 = -1.305038314 Simplifying t + -0.375 = -1.305038314 Reorder the terms: -0.375 + t = -1.305038314 Solving -0.375 + t = -1.305038314 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.375' to each side of the equation. -0.375 + 0.375 + t = -1.305038314 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + t = -1.305038314 + 0.375 t = -1.305038314 + 0.375 Combine like terms: -1.305038314 + 0.375 = -0.930038314 t = -0.930038314 Simplifying t = -0.930038314Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.680038314, -0.930038314}
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