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Simplifying 16t2 + 4t + -27 = 0 Reorder the terms: -27 + 4t + 16t2 = 0 Solving -27 + 4t + 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'. -1.6875 + 0.25t + t2 = 0 Move the constant term to the right: Add '1.6875' to each side of the equation. -1.6875 + 0.25t + 1.6875 + t2 = 0 + 1.6875 Reorder the terms: -1.6875 + 1.6875 + 0.25t + t2 = 0 + 1.6875 Combine like terms: -1.6875 + 1.6875 = 0.0000 0.0000 + 0.25t + t2 = 0 + 1.6875 0.25t + t2 = 0 + 1.6875 Combine like terms: 0 + 1.6875 = 1.6875 0.25t + t2 = 1.6875 The t term is 0.25t. Take half its coefficient (0.125). Square it (0.015625) and add it to both sides. Add '0.015625' to each side of the equation. 0.25t + 0.015625 + t2 = 1.6875 + 0.015625 Reorder the terms: 0.015625 + 0.25t + t2 = 1.6875 + 0.015625 Combine like terms: 1.6875 + 0.015625 = 1.703125 0.015625 + 0.25t + t2 = 1.703125 Factor a perfect square on the left side: (t + 0.125)(t + 0.125) = 1.703125 Calculate the square root of the right side: 1.305038314 Break this problem into two subproblems by setting (t + 0.125) equal to 1.305038314 and -1.305038314.Subproblem 1
t + 0.125 = 1.305038314 Simplifying t + 0.125 = 1.305038314 Reorder the terms: 0.125 + t = 1.305038314 Solving 0.125 + t = 1.305038314 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.125' to each side of the equation. 0.125 + -0.125 + t = 1.305038314 + -0.125 Combine like terms: 0.125 + -0.125 = 0.000 0.000 + t = 1.305038314 + -0.125 t = 1.305038314 + -0.125 Combine like terms: 1.305038314 + -0.125 = 1.180038314 t = 1.180038314 Simplifying t = 1.180038314Subproblem 2
t + 0.125 = -1.305038314 Simplifying t + 0.125 = -1.305038314 Reorder the terms: 0.125 + t = -1.305038314 Solving 0.125 + t = -1.305038314 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.125' to each side of the equation. 0.125 + -0.125 + t = -1.305038314 + -0.125 Combine like terms: 0.125 + -0.125 = 0.000 0.000 + t = -1.305038314 + -0.125 t = -1.305038314 + -0.125 Combine like terms: -1.305038314 + -0.125 = -1.430038314 t = -1.430038314 Simplifying t = -1.430038314Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.180038314, -1.430038314}
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