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Simplifying 40 = 9.8t2 + 140t + 1 Reorder the terms: 40 = 1 + 140t + 9.8t2 Solving 40 = 1 + 140t + 9.8t2 Solving for variable 't'. Combine like terms: 40 + -1 = 39 39 + -140t + -9.8t2 = 1 + 140t + 9.8t2 + -1 + -140t + -9.8t2 Reorder the terms: 39 + -140t + -9.8t2 = 1 + -1 + 140t + -140t + 9.8t2 + -9.8t2 Combine like terms: 1 + -1 = 0 39 + -140t + -9.8t2 = 0 + 140t + -140t + 9.8t2 + -9.8t2 39 + -140t + -9.8t2 = 140t + -140t + 9.8t2 + -9.8t2 Combine like terms: 140t + -140t = 0 39 + -140t + -9.8t2 = 0 + 9.8t2 + -9.8t2 39 + -140t + -9.8t2 = 9.8t2 + -9.8t2 Combine like terms: 9.8t2 + -9.8t2 = 0.0 39 + -140t + -9.8t2 = 0.0 Begin completing the square. Divide all terms by -9.8 the coefficient of the squared term: Divide each side by '-9.8'. -3.979591837 + 14.28571429t + t2 = 0 Move the constant term to the right: Add '3.979591837' to each side of the equation. -3.979591837 + 14.28571429t + 3.979591837 + t2 = 0 + 3.979591837 Reorder the terms: -3.979591837 + 3.979591837 + 14.28571429t + t2 = 0 + 3.979591837 Combine like terms: -3.979591837 + 3.979591837 = 0.000000000 0.000000000 + 14.28571429t + t2 = 0 + 3.979591837 14.28571429t + t2 = 0 + 3.979591837 Combine like terms: 0 + 3.979591837 = 3.979591837 14.28571429t + t2 = 3.979591837 The t term is 14.28571429t. Take half its coefficient (7.142857145). Square it (51.02040819) and add it to both sides. Add '51.02040819' to each side of the equation. 14.28571429t + 51.02040819 + t2 = 3.979591837 + 51.02040819 Reorder the terms: 51.02040819 + 14.28571429t + t2 = 3.979591837 + 51.02040819 Combine like terms: 3.979591837 + 51.02040819 = 55.000000027 51.02040819 + 14.28571429t + t2 = 55.000000027 Factor a perfect square on the left side: (t + 7.142857145)(t + 7.142857145) = 55.000000027 Calculate the square root of the right side: 7.416198489 Break this problem into two subproblems by setting (t + 7.142857145) equal to 7.416198489 and -7.416198489.Subproblem 1
t + 7.142857145 = 7.416198489 Simplifying t + 7.142857145 = 7.416198489 Reorder the terms: 7.142857145 + t = 7.416198489 Solving 7.142857145 + t = 7.416198489 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-7.142857145' to each side of the equation. 7.142857145 + -7.142857145 + t = 7.416198489 + -7.142857145 Combine like terms: 7.142857145 + -7.142857145 = 0.000000000 0.000000000 + t = 7.416198489 + -7.142857145 t = 7.416198489 + -7.142857145 Combine like terms: 7.416198489 + -7.142857145 = 0.273341344 t = 0.273341344 Simplifying t = 0.273341344Subproblem 2
t + 7.142857145 = -7.416198489 Simplifying t + 7.142857145 = -7.416198489 Reorder the terms: 7.142857145 + t = -7.416198489 Solving 7.142857145 + t = -7.416198489 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-7.142857145' to each side of the equation. 7.142857145 + -7.142857145 + t = -7.416198489 + -7.142857145 Combine like terms: 7.142857145 + -7.142857145 = 0.000000000 0.000000000 + t = -7.416198489 + -7.142857145 t = -7.416198489 + -7.142857145 Combine like terms: -7.416198489 + -7.142857145 = -14.559055634 t = -14.559055634 Simplifying t = -14.559055634Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.273341344, -14.559055634}
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