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Simplifying 0 = -4.9t2 + -6t + 500 Reorder the terms: 0 = 500 + -6t + -4.9t2 Solving 0 = 500 + -6t + -4.9t2 Solving for variable 't'. Combine like terms: 0 + -500 = -500 -500 + 6t + 4.9t2 = 500 + -6t + -4.9t2 + -500 + 6t + 4.9t2 Reorder the terms: -500 + 6t + 4.9t2 = 500 + -500 + -6t + 6t + -4.9t2 + 4.9t2 Combine like terms: 500 + -500 = 0 -500 + 6t + 4.9t2 = 0 + -6t + 6t + -4.9t2 + 4.9t2 -500 + 6t + 4.9t2 = -6t + 6t + -4.9t2 + 4.9t2 Combine like terms: -6t + 6t = 0 -500 + 6t + 4.9t2 = 0 + -4.9t2 + 4.9t2 -500 + 6t + 4.9t2 = -4.9t2 + 4.9t2 Combine like terms: -4.9t2 + 4.9t2 = 0.0 -500 + 6t + 4.9t2 = 0.0 Begin completing the square. Divide all terms by 4.9 the coefficient of the squared term: Divide each side by '4.9'. -102.0408163 + 1.224489796t + t2 = 0 Move the constant term to the right: Add '102.0408163' to each side of the equation. -102.0408163 + 1.224489796t + 102.0408163 + t2 = 0 + 102.0408163 Reorder the terms: -102.0408163 + 102.0408163 + 1.224489796t + t2 = 0 + 102.0408163 Combine like terms: -102.0408163 + 102.0408163 = 0.0000000 0.0000000 + 1.224489796t + t2 = 0 + 102.0408163 1.224489796t + t2 = 0 + 102.0408163 Combine like terms: 0 + 102.0408163 = 102.0408163 1.224489796t + t2 = 102.0408163 The t term is 1.224489796t. Take half its coefficient (0.612244898). Square it (0.3748438151) and add it to both sides. Add '0.3748438151' to each side of the equation. 1.224489796t + 0.3748438151 + t2 = 102.0408163 + 0.3748438151 Reorder the terms: 0.3748438151 + 1.224489796t + t2 = 102.0408163 + 0.3748438151 Combine like terms: 102.0408163 + 0.3748438151 = 102.4156601151 0.3748438151 + 1.224489796t + t2 = 102.4156601151 Factor a perfect square on the left side: (t + 0.612244898)(t + 0.612244898) = 102.4156601151 Calculate the square root of the right side: 10.120062258 Break this problem into two subproblems by setting (t + 0.612244898) equal to 10.120062258 and -10.120062258.Subproblem 1
t + 0.612244898 = 10.120062258 Simplifying t + 0.612244898 = 10.120062258 Reorder the terms: 0.612244898 + t = 10.120062258 Solving 0.612244898 + t = 10.120062258 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.612244898' to each side of the equation. 0.612244898 + -0.612244898 + t = 10.120062258 + -0.612244898 Combine like terms: 0.612244898 + -0.612244898 = 0.000000000 0.000000000 + t = 10.120062258 + -0.612244898 t = 10.120062258 + -0.612244898 Combine like terms: 10.120062258 + -0.612244898 = 9.50781736 t = 9.50781736 Simplifying t = 9.50781736Subproblem 2
t + 0.612244898 = -10.120062258 Simplifying t + 0.612244898 = -10.120062258 Reorder the terms: 0.612244898 + t = -10.120062258 Solving 0.612244898 + t = -10.120062258 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.612244898' to each side of the equation. 0.612244898 + -0.612244898 + t = -10.120062258 + -0.612244898 Combine like terms: 0.612244898 + -0.612244898 = 0.000000000 0.000000000 + t = -10.120062258 + -0.612244898 t = -10.120062258 + -0.612244898 Combine like terms: -10.120062258 + -0.612244898 = -10.732307156 t = -10.732307156 Simplifying t = -10.732307156Solution
The solution to the problem is based on the solutions from the subproblems. t = {9.50781736, -10.732307156}
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