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Simplifying 16t2 + 54t + 7 = 0 Reorder the terms: 7 + 54t + 16t2 = 0 Solving 7 + 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'. 0.4375 + 3.375t + t2 = 0 Move the constant term to the right: Add '-0.4375' to each side of the equation. 0.4375 + 3.375t + -0.4375 + t2 = 0 + -0.4375 Reorder the terms: 0.4375 + -0.4375 + 3.375t + t2 = 0 + -0.4375 Combine like terms: 0.4375 + -0.4375 = 0.0000 0.0000 + 3.375t + t2 = 0 + -0.4375 3.375t + t2 = 0 + -0.4375 Combine like terms: 0 + -0.4375 = -0.4375 3.375t + t2 = -0.4375 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 = -0.4375 + 2.84765625 Reorder the terms: 2.84765625 + 3.375t + t2 = -0.4375 + 2.84765625 Combine like terms: -0.4375 + 2.84765625 = 2.41015625 2.84765625 + 3.375t + t2 = 2.41015625 Factor a perfect square on the left side: (t + 1.6875)(t + 1.6875) = 2.41015625 Calculate the square root of the right side: 1.552467794 Break this problem into two subproblems by setting (t + 1.6875) equal to 1.552467794 and -1.552467794.Subproblem 1
t + 1.6875 = 1.552467794 Simplifying t + 1.6875 = 1.552467794 Reorder the terms: 1.6875 + t = 1.552467794 Solving 1.6875 + t = 1.552467794 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 = 1.552467794 + -1.6875 Combine like terms: 1.6875 + -1.6875 = 0.0000 0.0000 + t = 1.552467794 + -1.6875 t = 1.552467794 + -1.6875 Combine like terms: 1.552467794 + -1.6875 = -0.135032206 t = -0.135032206 Simplifying t = -0.135032206Subproblem 2
t + 1.6875 = -1.552467794 Simplifying t + 1.6875 = -1.552467794 Reorder the terms: 1.6875 + t = -1.552467794 Solving 1.6875 + t = -1.552467794 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 = -1.552467794 + -1.6875 Combine like terms: 1.6875 + -1.6875 = 0.0000 0.0000 + t = -1.552467794 + -1.6875 t = -1.552467794 + -1.6875 Combine like terms: -1.552467794 + -1.6875 = -3.239967794 t = -3.239967794 Simplifying t = -3.239967794Solution
The solution to the problem is based on the solutions from the subproblems. t = {-0.135032206, -3.239967794}
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