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Simplifying 0 = -16t2 + 27t + 16 Reorder the terms: 0 = 16 + 27t + -16t2 Solving 0 = 16 + 27t + -16t2 Solving for variable 't'. Combine like terms: 0 + -16 = -16 -16 + -27t + 16t2 = 16 + 27t + -16t2 + -16 + -27t + 16t2 Reorder the terms: -16 + -27t + 16t2 = 16 + -16 + 27t + -27t + -16t2 + 16t2 Combine like terms: 16 + -16 = 0 -16 + -27t + 16t2 = 0 + 27t + -27t + -16t2 + 16t2 -16 + -27t + 16t2 = 27t + -27t + -16t2 + 16t2 Combine like terms: 27t + -27t = 0 -16 + -27t + 16t2 = 0 + -16t2 + 16t2 -16 + -27t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -16 + -27t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -1 + -1.6875t + t2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + -1.6875t + 1 + t2 = 0 + 1 Reorder the terms: -1 + 1 + -1.6875t + t2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + -1.6875t + t2 = 0 + 1 -1.6875t + t2 = 0 + 1 Combine like terms: 0 + 1 = 1 -1.6875t + t2 = 1 The t term is -1.6875t. Take half its coefficient (-0.84375). Square it (0.7119140625) and add it to both sides. Add '0.7119140625' to each side of the equation. -1.6875t + 0.7119140625 + t2 = 1 + 0.7119140625 Reorder the terms: 0.7119140625 + -1.6875t + t2 = 1 + 0.7119140625 Combine like terms: 1 + 0.7119140625 = 1.7119140625 0.7119140625 + -1.6875t + t2 = 1.7119140625 Factor a perfect square on the left side: (t + -0.84375)(t + -0.84375) = 1.7119140625 Calculate the square root of the right side: 1.308401338 Break this problem into two subproblems by setting (t + -0.84375) equal to 1.308401338 and -1.308401338.Subproblem 1
t + -0.84375 = 1.308401338 Simplifying t + -0.84375 = 1.308401338 Reorder the terms: -0.84375 + t = 1.308401338 Solving -0.84375 + t = 1.308401338 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.84375' to each side of the equation. -0.84375 + 0.84375 + t = 1.308401338 + 0.84375 Combine like terms: -0.84375 + 0.84375 = 0.00000 0.00000 + t = 1.308401338 + 0.84375 t = 1.308401338 + 0.84375 Combine like terms: 1.308401338 + 0.84375 = 2.152151338 t = 2.152151338 Simplifying t = 2.152151338Subproblem 2
t + -0.84375 = -1.308401338 Simplifying t + -0.84375 = -1.308401338 Reorder the terms: -0.84375 + t = -1.308401338 Solving -0.84375 + t = -1.308401338 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.84375' to each side of the equation. -0.84375 + 0.84375 + t = -1.308401338 + 0.84375 Combine like terms: -0.84375 + 0.84375 = 0.00000 0.00000 + t = -1.308401338 + 0.84375 t = -1.308401338 + 0.84375 Combine like terms: -1.308401338 + 0.84375 = -0.464651338 t = -0.464651338 Simplifying t = -0.464651338Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.152151338, -0.464651338}
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