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Simplifying 190 = 19t2 + -100t Reorder the terms: 190 = -100t + 19t2 Solving 190 = -100t + 19t2 Solving for variable 't'. Reorder the terms: 190 + 100t + -19t2 = -100t + 100t + 19t2 + -19t2 Combine like terms: -100t + 100t = 0 190 + 100t + -19t2 = 0 + 19t2 + -19t2 190 + 100t + -19t2 = 19t2 + -19t2 Combine like terms: 19t2 + -19t2 = 0 190 + 100t + -19t2 = 0 Begin completing the square. Divide all terms by -19 the coefficient of the squared term: Divide each side by '-19'. -10 + -5.263157895t + t2 = 0 Move the constant term to the right: Add '10' to each side of the equation. -10 + -5.263157895t + 10 + t2 = 0 + 10 Reorder the terms: -10 + 10 + -5.263157895t + t2 = 0 + 10 Combine like terms: -10 + 10 = 0 0 + -5.263157895t + t2 = 0 + 10 -5.263157895t + t2 = 0 + 10 Combine like terms: 0 + 10 = 10 -5.263157895t + t2 = 10 The t term is -5.263157895t. Take half its coefficient (-2.631578948). Square it (6.925207760) and add it to both sides. Add '6.925207760' to each side of the equation. -5.263157895t + 6.925207760 + t2 = 10 + 6.925207760 Reorder the terms: 6.925207760 + -5.263157895t + t2 = 10 + 6.925207760 Combine like terms: 10 + 6.925207760 = 16.92520776 6.925207760 + -5.263157895t + t2 = 16.92520776 Factor a perfect square on the left side: (t + -2.631578948)(t + -2.631578948) = 16.92520776 Calculate the square root of the right side: 4.114025736 Break this problem into two subproblems by setting (t + -2.631578948) equal to 4.114025736 and -4.114025736.Subproblem 1
t + -2.631578948 = 4.114025736 Simplifying t + -2.631578948 = 4.114025736 Reorder the terms: -2.631578948 + t = 4.114025736 Solving -2.631578948 + t = 4.114025736 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.631578948' to each side of the equation. -2.631578948 + 2.631578948 + t = 4.114025736 + 2.631578948 Combine like terms: -2.631578948 + 2.631578948 = 0.000000000 0.000000000 + t = 4.114025736 + 2.631578948 t = 4.114025736 + 2.631578948 Combine like terms: 4.114025736 + 2.631578948 = 6.745604684 t = 6.745604684 Simplifying t = 6.745604684Subproblem 2
t + -2.631578948 = -4.114025736 Simplifying t + -2.631578948 = -4.114025736 Reorder the terms: -2.631578948 + t = -4.114025736 Solving -2.631578948 + t = -4.114025736 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.631578948' to each side of the equation. -2.631578948 + 2.631578948 + t = -4.114025736 + 2.631578948 Combine like terms: -2.631578948 + 2.631578948 = 0.000000000 0.000000000 + t = -4.114025736 + 2.631578948 t = -4.114025736 + 2.631578948 Combine like terms: -4.114025736 + 2.631578948 = -1.482446788 t = -1.482446788 Simplifying t = -1.482446788Solution
The solution to the problem is based on the solutions from the subproblems. t = {6.745604684, -1.482446788}
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