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Simplifying 0 = 3.5t2 + -24t + -140 Reorder the terms: 0 = -140 + -24t + 3.5t2 Solving 0 = -140 + -24t + 3.5t2 Solving for variable 't'. Combine like terms: 0 + 140 = 140 140 + 24t + -3.5t2 = -140 + -24t + 3.5t2 + 140 + 24t + -3.5t2 Reorder the terms: 140 + 24t + -3.5t2 = -140 + 140 + -24t + 24t + 3.5t2 + -3.5t2 Combine like terms: -140 + 140 = 0 140 + 24t + -3.5t2 = 0 + -24t + 24t + 3.5t2 + -3.5t2 140 + 24t + -3.5t2 = -24t + 24t + 3.5t2 + -3.5t2 Combine like terms: -24t + 24t = 0 140 + 24t + -3.5t2 = 0 + 3.5t2 + -3.5t2 140 + 24t + -3.5t2 = 3.5t2 + -3.5t2 Combine like terms: 3.5t2 + -3.5t2 = 0.0 140 + 24t + -3.5t2 = 0.0 Begin completing the square. Divide all terms by -3.5 the coefficient of the squared term: Divide each side by '-3.5'. -40 + -6.857142857t + t2 = 0 Move the constant term to the right: Add '40' to each side of the equation. -40 + -6.857142857t + 40 + t2 = 0 + 40 Reorder the terms: -40 + 40 + -6.857142857t + t2 = 0 + 40 Combine like terms: -40 + 40 = 0 0 + -6.857142857t + t2 = 0 + 40 -6.857142857t + t2 = 0 + 40 Combine like terms: 0 + 40 = 40 -6.857142857t + t2 = 40 The t term is -6.857142857t. Take half its coefficient (-3.428571429). Square it (11.75510204) and add it to both sides. Add '11.75510204' to each side of the equation. -6.857142857t + 11.75510204 + t2 = 40 + 11.75510204 Reorder the terms: 11.75510204 + -6.857142857t + t2 = 40 + 11.75510204 Combine like terms: 40 + 11.75510204 = 51.75510204 11.75510204 + -6.857142857t + t2 = 51.75510204 Factor a perfect square on the left side: (t + -3.428571429)(t + -3.428571429) = 51.75510204 Calculate the square root of the right side: 7.194101893 Break this problem into two subproblems by setting (t + -3.428571429) equal to 7.194101893 and -7.194101893.Subproblem 1
t + -3.428571429 = 7.194101893 Simplifying t + -3.428571429 = 7.194101893 Reorder the terms: -3.428571429 + t = 7.194101893 Solving -3.428571429 + t = 7.194101893 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3.428571429' to each side of the equation. -3.428571429 + 3.428571429 + t = 7.194101893 + 3.428571429 Combine like terms: -3.428571429 + 3.428571429 = 0.000000000 0.000000000 + t = 7.194101893 + 3.428571429 t = 7.194101893 + 3.428571429 Combine like terms: 7.194101893 + 3.428571429 = 10.622673322 t = 10.622673322 Simplifying t = 10.622673322Subproblem 2
t + -3.428571429 = -7.194101893 Simplifying t + -3.428571429 = -7.194101893 Reorder the terms: -3.428571429 + t = -7.194101893 Solving -3.428571429 + t = -7.194101893 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3.428571429' to each side of the equation. -3.428571429 + 3.428571429 + t = -7.194101893 + 3.428571429 Combine like terms: -3.428571429 + 3.428571429 = 0.000000000 0.000000000 + t = -7.194101893 + 3.428571429 t = -7.194101893 + 3.428571429 Combine like terms: -7.194101893 + 3.428571429 = -3.765530464 t = -3.765530464 Simplifying t = -3.765530464Solution
The solution to the problem is based on the solutions from the subproblems. t = {10.622673322, -3.765530464}
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