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Simplifying 65(t) = -16t2 + 85t + 1 Reorder the terms: 65t = 1 + 85t + -16t2 Solving 65t = 1 + 85t + -16t2 Solving for variable 't'. Reorder the terms: -1 + 65t + -85t + 16t2 = 1 + 85t + -16t2 + -1 + -85t + 16t2 Combine like terms: 65t + -85t = -20t -1 + -20t + 16t2 = 1 + 85t + -16t2 + -1 + -85t + 16t2 Reorder the terms: -1 + -20t + 16t2 = 1 + -1 + 85t + -85t + -16t2 + 16t2 Combine like terms: 1 + -1 = 0 -1 + -20t + 16t2 = 0 + 85t + -85t + -16t2 + 16t2 -1 + -20t + 16t2 = 85t + -85t + -16t2 + 16t2 Combine like terms: 85t + -85t = 0 -1 + -20t + 16t2 = 0 + -16t2 + 16t2 -1 + -20t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -1 + -20t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -0.0625 + -1.25t + t2 = 0 Move the constant term to the right: Add '0.0625' to each side of the equation. -0.0625 + -1.25t + 0.0625 + t2 = 0 + 0.0625 Reorder the terms: -0.0625 + 0.0625 + -1.25t + t2 = 0 + 0.0625 Combine like terms: -0.0625 + 0.0625 = 0.0000 0.0000 + -1.25t + t2 = 0 + 0.0625 -1.25t + t2 = 0 + 0.0625 Combine like terms: 0 + 0.0625 = 0.0625 -1.25t + t2 = 0.0625 The t term is -1.25t. Take half its coefficient (-0.625). Square it (0.390625) and add it to both sides. Add '0.390625' to each side of the equation. -1.25t + 0.390625 + t2 = 0.0625 + 0.390625 Reorder the terms: 0.390625 + -1.25t + t2 = 0.0625 + 0.390625 Combine like terms: 0.0625 + 0.390625 = 0.453125 0.390625 + -1.25t + t2 = 0.453125 Factor a perfect square on the left side: (t + -0.625)(t + -0.625) = 0.453125 Calculate the square root of the right side: 0.673145601 Break this problem into two subproblems by setting (t + -0.625) equal to 0.673145601 and -0.673145601.Subproblem 1
t + -0.625 = 0.673145601 Simplifying t + -0.625 = 0.673145601 Reorder the terms: -0.625 + t = 0.673145601 Solving -0.625 + t = 0.673145601 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.625' to each side of the equation. -0.625 + 0.625 + t = 0.673145601 + 0.625 Combine like terms: -0.625 + 0.625 = 0.000 0.000 + t = 0.673145601 + 0.625 t = 0.673145601 + 0.625 Combine like terms: 0.673145601 + 0.625 = 1.298145601 t = 1.298145601 Simplifying t = 1.298145601Subproblem 2
t + -0.625 = -0.673145601 Simplifying t + -0.625 = -0.673145601 Reorder the terms: -0.625 + t = -0.673145601 Solving -0.625 + t = -0.673145601 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.625' to each side of the equation. -0.625 + 0.625 + t = -0.673145601 + 0.625 Combine like terms: -0.625 + 0.625 = 0.000 0.000 + t = -0.673145601 + 0.625 t = -0.673145601 + 0.625 Combine like terms: -0.673145601 + 0.625 = -0.048145601 t = -0.048145601 Simplifying t = -0.048145601Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.298145601, -0.048145601}
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