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Simplifying t = -16t2 + 40t + 1.5 Reorder the terms: t = 1.5 + 40t + -16t2 Solving t = 1.5 + 40t + -16t2 Solving for variable 't'. Reorder the terms: -1.5 + t + -40t + 16t2 = 1.5 + 40t + -16t2 + -1.5 + -40t + 16t2 Combine like terms: t + -40t = -39t -1.5 + -39t + 16t2 = 1.5 + 40t + -16t2 + -1.5 + -40t + 16t2 Reorder the terms: -1.5 + -39t + 16t2 = 1.5 + -1.5 + 40t + -40t + -16t2 + 16t2 Combine like terms: 1.5 + -1.5 = 0.0 -1.5 + -39t + 16t2 = 0.0 + 40t + -40t + -16t2 + 16t2 -1.5 + -39t + 16t2 = 40t + -40t + -16t2 + 16t2 Combine like terms: 40t + -40t = 0 -1.5 + -39t + 16t2 = 0 + -16t2 + 16t2 -1.5 + -39t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -1.5 + -39t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -0.09375 + -2.4375t + t2 = 0 Move the constant term to the right: Add '0.09375' to each side of the equation. -0.09375 + -2.4375t + 0.09375 + t2 = 0 + 0.09375 Reorder the terms: -0.09375 + 0.09375 + -2.4375t + t2 = 0 + 0.09375 Combine like terms: -0.09375 + 0.09375 = 0.00000 0.00000 + -2.4375t + t2 = 0 + 0.09375 -2.4375t + t2 = 0 + 0.09375 Combine like terms: 0 + 0.09375 = 0.09375 -2.4375t + t2 = 0.09375 The t term is -2.4375t. Take half its coefficient (-1.21875). Square it (1.485351563) and add it to both sides. Add '1.485351563' to each side of the equation. -2.4375t + 1.485351563 + t2 = 0.09375 + 1.485351563 Reorder the terms: 1.485351563 + -2.4375t + t2 = 0.09375 + 1.485351563 Combine like terms: 0.09375 + 1.485351563 = 1.579101563 1.485351563 + -2.4375t + t2 = 1.579101563 Factor a perfect square on the left side: (t + -1.21875)(t + -1.21875) = 1.579101563 Calculate the square root of the right side: 1.256623079 Break this problem into two subproblems by setting (t + -1.21875) equal to 1.256623079 and -1.256623079.Subproblem 1
t + -1.21875 = 1.256623079 Simplifying t + -1.21875 = 1.256623079 Reorder the terms: -1.21875 + t = 1.256623079 Solving -1.21875 + t = 1.256623079 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.21875' to each side of the equation. -1.21875 + 1.21875 + t = 1.256623079 + 1.21875 Combine like terms: -1.21875 + 1.21875 = 0.00000 0.00000 + t = 1.256623079 + 1.21875 t = 1.256623079 + 1.21875 Combine like terms: 1.256623079 + 1.21875 = 2.475373079 t = 2.475373079 Simplifying t = 2.475373079Subproblem 2
t + -1.21875 = -1.256623079 Simplifying t + -1.21875 = -1.256623079 Reorder the terms: -1.21875 + t = -1.256623079 Solving -1.21875 + t = -1.256623079 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.21875' to each side of the equation. -1.21875 + 1.21875 + t = -1.256623079 + 1.21875 Combine like terms: -1.21875 + 1.21875 = 0.00000 0.00000 + t = -1.256623079 + 1.21875 t = -1.256623079 + 1.21875 Combine like terms: -1.256623079 + 1.21875 = -0.037873079 t = -0.037873079 Simplifying t = -0.037873079Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.475373079, -0.037873079}
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