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Simplifying t2 + 9 + -9t + -27 + 20 = 0 Reorder the terms: 9 + -27 + 20 + -9t + t2 = 0 Combine like terms: 9 + -27 = -18 -18 + 20 + -9t + t2 = 0 Combine like terms: -18 + 20 = 2 2 + -9t + t2 = 0 Solving 2 + -9t + t2 = 0 Solving for variable 't'. Begin completing the square. Move the constant term to the right: Add '-2' to each side of the equation. 2 + -9t + -2 + t2 = 0 + -2 Reorder the terms: 2 + -2 + -9t + t2 = 0 + -2 Combine like terms: 2 + -2 = 0 0 + -9t + t2 = 0 + -2 -9t + t2 = 0 + -2 Combine like terms: 0 + -2 = -2 -9t + t2 = -2 The t term is -9t. Take half its coefficient (-4.5). Square it (20.25) and add it to both sides. Add '20.25' to each side of the equation. -9t + 20.25 + t2 = -2 + 20.25 Reorder the terms: 20.25 + -9t + t2 = -2 + 20.25 Combine like terms: -2 + 20.25 = 18.25 20.25 + -9t + t2 = 18.25 Factor a perfect square on the left side: (t + -4.5)(t + -4.5) = 18.25 Calculate the square root of the right side: 4.272001873 Break this problem into two subproblems by setting (t + -4.5) equal to 4.272001873 and -4.272001873.Subproblem 1
t + -4.5 = 4.272001873 Simplifying t + -4.5 = 4.272001873 Reorder the terms: -4.5 + t = 4.272001873 Solving -4.5 + t = 4.272001873 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + t = 4.272001873 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + t = 4.272001873 + 4.5 t = 4.272001873 + 4.5 Combine like terms: 4.272001873 + 4.5 = 8.772001873 t = 8.772001873 Simplifying t = 8.772001873Subproblem 2
t + -4.5 = -4.272001873 Simplifying t + -4.5 = -4.272001873 Reorder the terms: -4.5 + t = -4.272001873 Solving -4.5 + t = -4.272001873 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + t = -4.272001873 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + t = -4.272001873 + 4.5 t = -4.272001873 + 4.5 Combine like terms: -4.272001873 + 4.5 = 0.227998127 t = 0.227998127 Simplifying t = 0.227998127Solution
The solution to the problem is based on the solutions from the subproblems. t = {8.772001873, 0.227998127}
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