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Simplifying 32.5 = 65t + -16t2 Solving 32.5 = 65t + -16t2 Solving for variable 't'. Reorder the terms: 32.5 + -65t + 16t2 = 65t + -65t + -16t2 + 16t2 Combine like terms: 65t + -65t = 0 32.5 + -65t + 16t2 = 0 + -16t2 + 16t2 32.5 + -65t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 32.5 + -65t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 2.03125 + -4.0625t + t2 = 0 Move the constant term to the right: Add '-2.03125' to each side of the equation. 2.03125 + -4.0625t + -2.03125 + t2 = 0 + -2.03125 Reorder the terms: 2.03125 + -2.03125 + -4.0625t + t2 = 0 + -2.03125 Combine like terms: 2.03125 + -2.03125 = 0.00000 0.00000 + -4.0625t + t2 = 0 + -2.03125 -4.0625t + t2 = 0 + -2.03125 Combine like terms: 0 + -2.03125 = -2.03125 -4.0625t + t2 = -2.03125 The t term is -4.0625t. Take half its coefficient (-2.03125). Square it (4.125976563) and add it to both sides. Add '4.125976563' to each side of the equation. -4.0625t + 4.125976563 + t2 = -2.03125 + 4.125976563 Reorder the terms: 4.125976563 + -4.0625t + t2 = -2.03125 + 4.125976563 Combine like terms: -2.03125 + 4.125976563 = 2.094726563 4.125976563 + -4.0625t + t2 = 2.094726563 Factor a perfect square on the left side: (t + -2.03125)(t + -2.03125) = 2.094726563 Calculate the square root of the right side: 1.447317022 Break this problem into two subproblems by setting (t + -2.03125) equal to 1.447317022 and -1.447317022.Subproblem 1
t + -2.03125 = 1.447317022 Simplifying t + -2.03125 = 1.447317022 Reorder the terms: -2.03125 + t = 1.447317022 Solving -2.03125 + t = 1.447317022 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.03125' to each side of the equation. -2.03125 + 2.03125 + t = 1.447317022 + 2.03125 Combine like terms: -2.03125 + 2.03125 = 0.00000 0.00000 + t = 1.447317022 + 2.03125 t = 1.447317022 + 2.03125 Combine like terms: 1.447317022 + 2.03125 = 3.478567022 t = 3.478567022 Simplifying t = 3.478567022Subproblem 2
t + -2.03125 = -1.447317022 Simplifying t + -2.03125 = -1.447317022 Reorder the terms: -2.03125 + t = -1.447317022 Solving -2.03125 + t = -1.447317022 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.03125' to each side of the equation. -2.03125 + 2.03125 + t = -1.447317022 + 2.03125 Combine like terms: -2.03125 + 2.03125 = 0.00000 0.00000 + t = -1.447317022 + 2.03125 t = -1.447317022 + 2.03125 Combine like terms: -1.447317022 + 2.03125 = 0.583932978 t = 0.583932978 Simplifying t = 0.583932978Solution
The solution to the problem is based on the solutions from the subproblems. t = {3.478567022, 0.583932978}
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