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Simplifying t2 + 27t + 75 = 0 Reorder the terms: 75 + 27t + t2 = 0 Solving 75 + 27t + t2 = 0 Solving for variable 't'. Begin completing the square. Move the constant term to the right: Add '-75' to each side of the equation. 75 + 27t + -75 + t2 = 0 + -75 Reorder the terms: 75 + -75 + 27t + t2 = 0 + -75 Combine like terms: 75 + -75 = 0 0 + 27t + t2 = 0 + -75 27t + t2 = 0 + -75 Combine like terms: 0 + -75 = -75 27t + t2 = -75 The t term is 27t. Take half its coefficient (13.5). Square it (182.25) and add it to both sides. Add '182.25' to each side of the equation. 27t + 182.25 + t2 = -75 + 182.25 Reorder the terms: 182.25 + 27t + t2 = -75 + 182.25 Combine like terms: -75 + 182.25 = 107.25 182.25 + 27t + t2 = 107.25 Factor a perfect square on the left side: (t + 13.5)(t + 13.5) = 107.25 Calculate the square root of the right side: 10.356157589 Break this problem into two subproblems by setting (t + 13.5) equal to 10.356157589 and -10.356157589.Subproblem 1
t + 13.5 = 10.356157589 Simplifying t + 13.5 = 10.356157589 Reorder the terms: 13.5 + t = 10.356157589 Solving 13.5 + t = 10.356157589 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-13.5' to each side of the equation. 13.5 + -13.5 + t = 10.356157589 + -13.5 Combine like terms: 13.5 + -13.5 = 0.0 0.0 + t = 10.356157589 + -13.5 t = 10.356157589 + -13.5 Combine like terms: 10.356157589 + -13.5 = -3.143842411 t = -3.143842411 Simplifying t = -3.143842411Subproblem 2
t + 13.5 = -10.356157589 Simplifying t + 13.5 = -10.356157589 Reorder the terms: 13.5 + t = -10.356157589 Solving 13.5 + t = -10.356157589 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-13.5' to each side of the equation. 13.5 + -13.5 + t = -10.356157589 + -13.5 Combine like terms: 13.5 + -13.5 = 0.0 0.0 + t = -10.356157589 + -13.5 t = -10.356157589 + -13.5 Combine like terms: -10.356157589 + -13.5 = -23.856157589 t = -23.856157589 Simplifying t = -23.856157589Solution
The solution to the problem is based on the solutions from the subproblems. t = {-3.143842411, -23.856157589}
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