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Simplifying 0 = -16t2 + 13t + 1 Reorder the terms: 0 = 1 + 13t + -16t2 Solving 0 = 1 + 13t + -16t2 Solving for variable 't'. Combine like terms: 0 + -1 = -1 -1 + -13t + 16t2 = 1 + 13t + -16t2 + -1 + -13t + 16t2 Reorder the terms: -1 + -13t + 16t2 = 1 + -1 + 13t + -13t + -16t2 + 16t2 Combine like terms: 1 + -1 = 0 -1 + -13t + 16t2 = 0 + 13t + -13t + -16t2 + 16t2 -1 + -13t + 16t2 = 13t + -13t + -16t2 + 16t2 Combine like terms: 13t + -13t = 0 -1 + -13t + 16t2 = 0 + -16t2 + 16t2 -1 + -13t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -1 + -13t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -0.0625 + -0.8125t + t2 = 0 Move the constant term to the right: Add '0.0625' to each side of the equation. -0.0625 + -0.8125t + 0.0625 + t2 = 0 + 0.0625 Reorder the terms: -0.0625 + 0.0625 + -0.8125t + t2 = 0 + 0.0625 Combine like terms: -0.0625 + 0.0625 = 0.0000 0.0000 + -0.8125t + t2 = 0 + 0.0625 -0.8125t + t2 = 0 + 0.0625 Combine like terms: 0 + 0.0625 = 0.0625 -0.8125t + t2 = 0.0625 The t term is -0.8125t. Take half its coefficient (-0.40625). Square it (0.1650390625) and add it to both sides. Add '0.1650390625' to each side of the equation. -0.8125t + 0.1650390625 + t2 = 0.0625 + 0.1650390625 Reorder the terms: 0.1650390625 + -0.8125t + t2 = 0.0625 + 0.1650390625 Combine like terms: 0.0625 + 0.1650390625 = 0.2275390625 0.1650390625 + -0.8125t + t2 = 0.2275390625 Factor a perfect square on the left side: (t + -0.40625)(t + -0.40625) = 0.2275390625 Calculate the square root of the right side: 0.477010548 Break this problem into two subproblems by setting (t + -0.40625) equal to 0.477010548 and -0.477010548.Subproblem 1
t + -0.40625 = 0.477010548 Simplifying t + -0.40625 = 0.477010548 Reorder the terms: -0.40625 + t = 0.477010548 Solving -0.40625 + t = 0.477010548 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.40625' to each side of the equation. -0.40625 + 0.40625 + t = 0.477010548 + 0.40625 Combine like terms: -0.40625 + 0.40625 = 0.00000 0.00000 + t = 0.477010548 + 0.40625 t = 0.477010548 + 0.40625 Combine like terms: 0.477010548 + 0.40625 = 0.883260548 t = 0.883260548 Simplifying t = 0.883260548Subproblem 2
t + -0.40625 = -0.477010548 Simplifying t + -0.40625 = -0.477010548 Reorder the terms: -0.40625 + t = -0.477010548 Solving -0.40625 + t = -0.477010548 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.40625' to each side of the equation. -0.40625 + 0.40625 + t = -0.477010548 + 0.40625 Combine like terms: -0.40625 + 0.40625 = 0.00000 0.00000 + t = -0.477010548 + 0.40625 t = -0.477010548 + 0.40625 Combine like terms: -0.477010548 + 0.40625 = -0.070760548 t = -0.070760548 Simplifying t = -0.070760548Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.883260548, -0.070760548}
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