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Simplifying -16t2 + 32t = -5 Reorder the terms: 32t + -16t2 = -5 Solving 32t + -16t2 = -5 Solving for variable 't'. Reorder the terms: 5 + 32t + -16t2 = -5 + 5 Combine like terms: -5 + 5 = 0 5 + 32t + -16t2 = 0 Begin completing the square. Divide all terms by -16 the coefficient of the squared term: Divide each side by '-16'. -0.3125 + -2t + t2 = 0 Move the constant term to the right: Add '0.3125' to each side of the equation. -0.3125 + -2t + 0.3125 + t2 = 0 + 0.3125 Reorder the terms: -0.3125 + 0.3125 + -2t + t2 = 0 + 0.3125 Combine like terms: -0.3125 + 0.3125 = 0.0000 0.0000 + -2t + t2 = 0 + 0.3125 -2t + t2 = 0 + 0.3125 Combine like terms: 0 + 0.3125 = 0.3125 -2t + t2 = 0.3125 The t term is -2t. Take half its coefficient (-1). Square it (1) and add it to both sides. Add '1' to each side of the equation. -2t + 1 + t2 = 0.3125 + 1 Reorder the terms: 1 + -2t + t2 = 0.3125 + 1 Combine like terms: 0.3125 + 1 = 1.3125 1 + -2t + t2 = 1.3125 Factor a perfect square on the left side: (t + -1)(t + -1) = 1.3125 Calculate the square root of the right side: 1.145643924 Break this problem into two subproblems by setting (t + -1) equal to 1.145643924 and -1.145643924.Subproblem 1
t + -1 = 1.145643924 Simplifying t + -1 = 1.145643924 Reorder the terms: -1 + t = 1.145643924 Solving -1 + t = 1.145643924 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + t = 1.145643924 + 1 Combine like terms: -1 + 1 = 0 0 + t = 1.145643924 + 1 t = 1.145643924 + 1 Combine like terms: 1.145643924 + 1 = 2.145643924 t = 2.145643924 Simplifying t = 2.145643924Subproblem 2
t + -1 = -1.145643924 Simplifying t + -1 = -1.145643924 Reorder the terms: -1 + t = -1.145643924 Solving -1 + t = -1.145643924 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + t = -1.145643924 + 1 Combine like terms: -1 + 1 = 0 0 + t = -1.145643924 + 1 t = -1.145643924 + 1 Combine like terms: -1.145643924 + 1 = -0.145643924 t = -0.145643924 Simplifying t = -0.145643924Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.145643924, -0.145643924}
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