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Simplifying -16t2 + 54t + 5 = 0 Reorder the terms: 5 + 54t + -16t2 = 0 Solving 5 + 54t + -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'. -0.3125 + -3.375t + t2 = 0 Move the constant term to the right: Add '0.3125' to each side of the equation. -0.3125 + -3.375t + 0.3125 + t2 = 0 + 0.3125 Reorder the terms: -0.3125 + 0.3125 + -3.375t + t2 = 0 + 0.3125 Combine like terms: -0.3125 + 0.3125 = 0.0000 0.0000 + -3.375t + t2 = 0 + 0.3125 -3.375t + t2 = 0 + 0.3125 Combine like terms: 0 + 0.3125 = 0.3125 -3.375t + t2 = 0.3125 The t term is -3.375t. Take half its coefficient (-1.6875). Square it (2.84765625) and add it to both sides. Add '2.84765625' to each side of the equation. -3.375t + 2.84765625 + t2 = 0.3125 + 2.84765625 Reorder the terms: 2.84765625 + -3.375t + t2 = 0.3125 + 2.84765625 Combine like terms: 0.3125 + 2.84765625 = 3.16015625 2.84765625 + -3.375t + t2 = 3.16015625 Factor a perfect square on the left side: (t + -1.6875)(t + -1.6875) = 3.16015625 Calculate the square root of the right side: 1.777682832 Break this problem into two subproblems by setting (t + -1.6875) equal to 1.777682832 and -1.777682832.Subproblem 1
t + -1.6875 = 1.777682832 Simplifying t + -1.6875 = 1.777682832 Reorder the terms: -1.6875 + t = 1.777682832 Solving -1.6875 + t = 1.777682832 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.6875' to each side of the equation. -1.6875 + 1.6875 + t = 1.777682832 + 1.6875 Combine like terms: -1.6875 + 1.6875 = 0.0000 0.0000 + t = 1.777682832 + 1.6875 t = 1.777682832 + 1.6875 Combine like terms: 1.777682832 + 1.6875 = 3.465182832 t = 3.465182832 Simplifying t = 3.465182832Subproblem 2
t + -1.6875 = -1.777682832 Simplifying t + -1.6875 = -1.777682832 Reorder the terms: -1.6875 + t = -1.777682832 Solving -1.6875 + t = -1.777682832 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.6875' to each side of the equation. -1.6875 + 1.6875 + t = -1.777682832 + 1.6875 Combine like terms: -1.6875 + 1.6875 = 0.0000 0.0000 + t = -1.777682832 + 1.6875 t = -1.777682832 + 1.6875 Combine like terms: -1.777682832 + 1.6875 = -0.090182832 t = -0.090182832 Simplifying t = -0.090182832Solution
The solution to the problem is based on the solutions from the subproblems. t = {3.465182832, -0.090182832}
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