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Simplifying 2t2 + -39t + 105 = 0 Reorder the terms: 105 + -39t + 2t2 = 0 Solving 105 + -39t + 2t2 = 0 Solving for variable 't'. Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 52.5 + -19.5t + t2 = 0 Move the constant term to the right: Add '-52.5' to each side of the equation. 52.5 + -19.5t + -52.5 + t2 = 0 + -52.5 Reorder the terms: 52.5 + -52.5 + -19.5t + t2 = 0 + -52.5 Combine like terms: 52.5 + -52.5 = 0.0 0.0 + -19.5t + t2 = 0 + -52.5 -19.5t + t2 = 0 + -52.5 Combine like terms: 0 + -52.5 = -52.5 -19.5t + t2 = -52.5 The t term is -19.5t. Take half its coefficient (-9.75). Square it (95.0625) and add it to both sides. Add '95.0625' to each side of the equation. -19.5t + 95.0625 + t2 = -52.5 + 95.0625 Reorder the terms: 95.0625 + -19.5t + t2 = -52.5 + 95.0625 Combine like terms: -52.5 + 95.0625 = 42.5625 95.0625 + -19.5t + t2 = 42.5625 Factor a perfect square on the left side: (t + -9.75)(t + -9.75) = 42.5625 Calculate the square root of the right side: 6.523994175 Break this problem into two subproblems by setting (t + -9.75) equal to 6.523994175 and -6.523994175.Subproblem 1
t + -9.75 = 6.523994175 Simplifying t + -9.75 = 6.523994175 Reorder the terms: -9.75 + t = 6.523994175 Solving -9.75 + t = 6.523994175 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '9.75' to each side of the equation. -9.75 + 9.75 + t = 6.523994175 + 9.75 Combine like terms: -9.75 + 9.75 = 0.00 0.00 + t = 6.523994175 + 9.75 t = 6.523994175 + 9.75 Combine like terms: 6.523994175 + 9.75 = 16.273994175 t = 16.273994175 Simplifying t = 16.273994175Subproblem 2
t + -9.75 = -6.523994175 Simplifying t + -9.75 = -6.523994175 Reorder the terms: -9.75 + t = -6.523994175 Solving -9.75 + t = -6.523994175 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '9.75' to each side of the equation. -9.75 + 9.75 + t = -6.523994175 + 9.75 Combine like terms: -9.75 + 9.75 = 0.00 0.00 + t = -6.523994175 + 9.75 t = -6.523994175 + 9.75 Combine like terms: -6.523994175 + 9.75 = 3.226005825 t = 3.226005825 Simplifying t = 3.226005825Solution
The solution to the problem is based on the solutions from the subproblems. t = {16.273994175, 3.226005825}
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