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Simplifying 0 = -16t2 + -8t + 69 Reorder the terms: 0 = 69 + -8t + -16t2 Solving 0 = 69 + -8t + -16t2 Solving for variable 't'. Combine like terms: 0 + -69 = -69 -69 + 8t + 16t2 = 69 + -8t + -16t2 + -69 + 8t + 16t2 Reorder the terms: -69 + 8t + 16t2 = 69 + -69 + -8t + 8t + -16t2 + 16t2 Combine like terms: 69 + -69 = 0 -69 + 8t + 16t2 = 0 + -8t + 8t + -16t2 + 16t2 -69 + 8t + 16t2 = -8t + 8t + -16t2 + 16t2 Combine like terms: -8t + 8t = 0 -69 + 8t + 16t2 = 0 + -16t2 + 16t2 -69 + 8t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -69 + 8t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -4.3125 + 0.5t + t2 = 0 Move the constant term to the right: Add '4.3125' to each side of the equation. -4.3125 + 0.5t + 4.3125 + t2 = 0 + 4.3125 Reorder the terms: -4.3125 + 4.3125 + 0.5t + t2 = 0 + 4.3125 Combine like terms: -4.3125 + 4.3125 = 0.0000 0.0000 + 0.5t + t2 = 0 + 4.3125 0.5t + t2 = 0 + 4.3125 Combine like terms: 0 + 4.3125 = 4.3125 0.5t + t2 = 4.3125 The t term is 0.5t. Take half its coefficient (0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. 0.5t + 0.0625 + t2 = 4.3125 + 0.0625 Reorder the terms: 0.0625 + 0.5t + t2 = 4.3125 + 0.0625 Combine like terms: 4.3125 + 0.0625 = 4.375 0.0625 + 0.5t + t2 = 4.375 Factor a perfect square on the left side: (t + 0.25)(t + 0.25) = 4.375 Calculate the square root of the right side: 2.091650066 Break this problem into two subproblems by setting (t + 0.25) equal to 2.091650066 and -2.091650066.Subproblem 1
t + 0.25 = 2.091650066 Simplifying t + 0.25 = 2.091650066 Reorder the terms: 0.25 + t = 2.091650066 Solving 0.25 + t = 2.091650066 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + t = 2.091650066 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + t = 2.091650066 + -0.25 t = 2.091650066 + -0.25 Combine like terms: 2.091650066 + -0.25 = 1.841650066 t = 1.841650066 Simplifying t = 1.841650066Subproblem 2
t + 0.25 = -2.091650066 Simplifying t + 0.25 = -2.091650066 Reorder the terms: 0.25 + t = -2.091650066 Solving 0.25 + t = -2.091650066 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + t = -2.091650066 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + t = -2.091650066 + -0.25 t = -2.091650066 + -0.25 Combine like terms: -2.091650066 + -0.25 = -2.341650066 t = -2.341650066 Simplifying t = -2.341650066Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.841650066, -2.341650066}
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