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Simplifying 16t2 + 7t + -54 = 0 Reorder the terms: -54 + 7t + 16t2 = 0 Solving -54 + 7t + 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'. -3.375 + 0.4375t + t2 = 0 Move the constant term to the right: Add '3.375' to each side of the equation. -3.375 + 0.4375t + 3.375 + t2 = 0 + 3.375 Reorder the terms: -3.375 + 3.375 + 0.4375t + t2 = 0 + 3.375 Combine like terms: -3.375 + 3.375 = 0.000 0.000 + 0.4375t + t2 = 0 + 3.375 0.4375t + t2 = 0 + 3.375 Combine like terms: 0 + 3.375 = 3.375 0.4375t + t2 = 3.375 The t term is 0.4375t. Take half its coefficient (0.21875). Square it (0.0478515625) and add it to both sides. Add '0.0478515625' to each side of the equation. 0.4375t + 0.0478515625 + t2 = 3.375 + 0.0478515625 Reorder the terms: 0.0478515625 + 0.4375t + t2 = 3.375 + 0.0478515625 Combine like terms: 3.375 + 0.0478515625 = 3.4228515625 0.0478515625 + 0.4375t + t2 = 3.4228515625 Factor a perfect square on the left side: (t + 0.21875)(t + 0.21875) = 3.4228515625 Calculate the square root of the right side: 1.850095014 Break this problem into two subproblems by setting (t + 0.21875) equal to 1.850095014 and -1.850095014.Subproblem 1
t + 0.21875 = 1.850095014 Simplifying t + 0.21875 = 1.850095014 Reorder the terms: 0.21875 + t = 1.850095014 Solving 0.21875 + t = 1.850095014 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.21875' to each side of the equation. 0.21875 + -0.21875 + t = 1.850095014 + -0.21875 Combine like terms: 0.21875 + -0.21875 = 0.00000 0.00000 + t = 1.850095014 + -0.21875 t = 1.850095014 + -0.21875 Combine like terms: 1.850095014 + -0.21875 = 1.631345014 t = 1.631345014 Simplifying t = 1.631345014Subproblem 2
t + 0.21875 = -1.850095014 Simplifying t + 0.21875 = -1.850095014 Reorder the terms: 0.21875 + t = -1.850095014 Solving 0.21875 + t = -1.850095014 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.21875' to each side of the equation. 0.21875 + -0.21875 + t = -1.850095014 + -0.21875 Combine like terms: 0.21875 + -0.21875 = 0.00000 0.00000 + t = -1.850095014 + -0.21875 t = -1.850095014 + -0.21875 Combine like terms: -1.850095014 + -0.21875 = -2.068845014 t = -2.068845014 Simplifying t = -2.068845014Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.631345014, -2.068845014}
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