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Simplifying 16t2 + 60t + 5 = 0 Reorder the terms: 5 + 60t + 16t2 = 0 Solving 5 + 60t + 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 + 3.75t + t2 = 0 Move the constant term to the right: Add '-0.3125' to each side of the equation. 0.3125 + 3.75t + -0.3125 + t2 = 0 + -0.3125 Reorder the terms: 0.3125 + -0.3125 + 3.75t + t2 = 0 + -0.3125 Combine like terms: 0.3125 + -0.3125 = 0.0000 0.0000 + 3.75t + t2 = 0 + -0.3125 3.75t + t2 = 0 + -0.3125 Combine like terms: 0 + -0.3125 = -0.3125 3.75t + t2 = -0.3125 The t term is 3.75t. Take half its coefficient (1.875). Square it (3.515625) and add it to both sides. Add '3.515625' to each side of the equation. 3.75t + 3.515625 + t2 = -0.3125 + 3.515625 Reorder the terms: 3.515625 + 3.75t + t2 = -0.3125 + 3.515625 Combine like terms: -0.3125 + 3.515625 = 3.203125 3.515625 + 3.75t + t2 = 3.203125 Factor a perfect square on the left side: (t + 1.875)(t + 1.875) = 3.203125 Calculate the square root of the right side: 1.789727633 Break this problem into two subproblems by setting (t + 1.875) equal to 1.789727633 and -1.789727633.Subproblem 1
t + 1.875 = 1.789727633 Simplifying t + 1.875 = 1.789727633 Reorder the terms: 1.875 + t = 1.789727633 Solving 1.875 + t = 1.789727633 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1.875' to each side of the equation. 1.875 + -1.875 + t = 1.789727633 + -1.875 Combine like terms: 1.875 + -1.875 = 0.000 0.000 + t = 1.789727633 + -1.875 t = 1.789727633 + -1.875 Combine like terms: 1.789727633 + -1.875 = -0.085272367 t = -0.085272367 Simplifying t = -0.085272367Subproblem 2
t + 1.875 = -1.789727633 Simplifying t + 1.875 = -1.789727633 Reorder the terms: 1.875 + t = -1.789727633 Solving 1.875 + t = -1.789727633 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1.875' to each side of the equation. 1.875 + -1.875 + t = -1.789727633 + -1.875 Combine like terms: 1.875 + -1.875 = 0.000 0.000 + t = -1.789727633 + -1.875 t = -1.789727633 + -1.875 Combine like terms: -1.789727633 + -1.875 = -3.664727633 t = -3.664727633 Simplifying t = -3.664727633Solution
The solution to the problem is based on the solutions from the subproblems. t = {-0.085272367, -3.664727633}
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