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Simplifying 0 = -16t2 + 38t + 15 Reorder the terms: 0 = 15 + 38t + -16t2 Solving 0 = 15 + 38t + -16t2 Solving for variable 't'. Combine like terms: 0 + -15 = -15 -15 + -38t + 16t2 = 15 + 38t + -16t2 + -15 + -38t + 16t2 Reorder the terms: -15 + -38t + 16t2 = 15 + -15 + 38t + -38t + -16t2 + 16t2 Combine like terms: 15 + -15 = 0 -15 + -38t + 16t2 = 0 + 38t + -38t + -16t2 + 16t2 -15 + -38t + 16t2 = 38t + -38t + -16t2 + 16t2 Combine like terms: 38t + -38t = 0 -15 + -38t + 16t2 = 0 + -16t2 + 16t2 -15 + -38t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -15 + -38t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -0.9375 + -2.375t + t2 = 0 Move the constant term to the right: Add '0.9375' to each side of the equation. -0.9375 + -2.375t + 0.9375 + t2 = 0 + 0.9375 Reorder the terms: -0.9375 + 0.9375 + -2.375t + t2 = 0 + 0.9375 Combine like terms: -0.9375 + 0.9375 = 0.0000 0.0000 + -2.375t + t2 = 0 + 0.9375 -2.375t + t2 = 0 + 0.9375 Combine like terms: 0 + 0.9375 = 0.9375 -2.375t + t2 = 0.9375 The t term is -2.375t. Take half its coefficient (-1.1875). Square it (1.41015625) and add it to both sides. Add '1.41015625' to each side of the equation. -2.375t + 1.41015625 + t2 = 0.9375 + 1.41015625 Reorder the terms: 1.41015625 + -2.375t + t2 = 0.9375 + 1.41015625 Combine like terms: 0.9375 + 1.41015625 = 2.34765625 1.41015625 + -2.375t + t2 = 2.34765625 Factor a perfect square on the left side: (t + -1.1875)(t + -1.1875) = 2.34765625 Calculate the square root of the right side: 1.532206334 Break this problem into two subproblems by setting (t + -1.1875) equal to 1.532206334 and -1.532206334.Subproblem 1
t + -1.1875 = 1.532206334 Simplifying t + -1.1875 = 1.532206334 Reorder the terms: -1.1875 + t = 1.532206334 Solving -1.1875 + t = 1.532206334 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.1875' to each side of the equation. -1.1875 + 1.1875 + t = 1.532206334 + 1.1875 Combine like terms: -1.1875 + 1.1875 = 0.0000 0.0000 + t = 1.532206334 + 1.1875 t = 1.532206334 + 1.1875 Combine like terms: 1.532206334 + 1.1875 = 2.719706334 t = 2.719706334 Simplifying t = 2.719706334Subproblem 2
t + -1.1875 = -1.532206334 Simplifying t + -1.1875 = -1.532206334 Reorder the terms: -1.1875 + t = -1.532206334 Solving -1.1875 + t = -1.532206334 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.1875' to each side of the equation. -1.1875 + 1.1875 + t = -1.532206334 + 1.1875 Combine like terms: -1.1875 + 1.1875 = 0.0000 0.0000 + t = -1.532206334 + 1.1875 t = -1.532206334 + 1.1875 Combine like terms: -1.532206334 + 1.1875 = -0.344706334 t = -0.344706334 Simplifying t = -0.344706334Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.719706334, -0.344706334}
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