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Simplifying 65 = -16t2 + 80t Reorder the terms: 65 = 80t + -16t2 Solving 65 = 80t + -16t2 Solving for variable 't'. Reorder the terms: 65 + -80t + 16t2 = 80t + -80t + -16t2 + 16t2 Combine like terms: 80t + -80t = 0 65 + -80t + 16t2 = 0 + -16t2 + 16t2 65 + -80t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 65 + -80t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 4.0625 + -5t + t2 = 0 Move the constant term to the right: Add '-4.0625' to each side of the equation. 4.0625 + -5t + -4.0625 + t2 = 0 + -4.0625 Reorder the terms: 4.0625 + -4.0625 + -5t + t2 = 0 + -4.0625 Combine like terms: 4.0625 + -4.0625 = 0.0000 0.0000 + -5t + t2 = 0 + -4.0625 -5t + t2 = 0 + -4.0625 Combine like terms: 0 + -4.0625 = -4.0625 -5t + t2 = -4.0625 The t term is -5t. Take half its coefficient (-2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. -5t + 6.25 + t2 = -4.0625 + 6.25 Reorder the terms: 6.25 + -5t + t2 = -4.0625 + 6.25 Combine like terms: -4.0625 + 6.25 = 2.1875 6.25 + -5t + t2 = 2.1875 Factor a perfect square on the left side: (t + -2.5)(t + -2.5) = 2.1875 Calculate the square root of the right side: 1.479019946 Break this problem into two subproblems by setting (t + -2.5) equal to 1.479019946 and -1.479019946.Subproblem 1
t + -2.5 = 1.479019946 Simplifying t + -2.5 = 1.479019946 Reorder the terms: -2.5 + t = 1.479019946 Solving -2.5 + t = 1.479019946 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + t = 1.479019946 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + t = 1.479019946 + 2.5 t = 1.479019946 + 2.5 Combine like terms: 1.479019946 + 2.5 = 3.979019946 t = 3.979019946 Simplifying t = 3.979019946Subproblem 2
t + -2.5 = -1.479019946 Simplifying t + -2.5 = -1.479019946 Reorder the terms: -2.5 + t = -1.479019946 Solving -2.5 + t = -1.479019946 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + t = -1.479019946 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + t = -1.479019946 + 2.5 t = -1.479019946 + 2.5 Combine like terms: -1.479019946 + 2.5 = 1.020980054 t = 1.020980054 Simplifying t = 1.020980054Solution
The solution to the problem is based on the solutions from the subproblems. t = {3.979019946, 1.020980054}
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