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Simplifying -16t2 + 40t + -3 = 8 Reorder the terms: -3 + 40t + -16t2 = 8 Solving -3 + 40t + -16t2 = 8 Solving for variable 't'. Reorder the terms: -3 + -8 + 40t + -16t2 = 8 + -8 Combine like terms: -3 + -8 = -11 -11 + 40t + -16t2 = 8 + -8 Combine like terms: 8 + -8 = 0 -11 + 40t + -16t2 = 0 Begin completing the square. Divide all terms by -16 the coefficient of the squared term: Divide each side by '-16'. 0.6875 + -2.5t + t2 = 0 Move the constant term to the right: Add '-0.6875' to each side of the equation. 0.6875 + -2.5t + -0.6875 + t2 = 0 + -0.6875 Reorder the terms: 0.6875 + -0.6875 + -2.5t + t2 = 0 + -0.6875 Combine like terms: 0.6875 + -0.6875 = 0.0000 0.0000 + -2.5t + t2 = 0 + -0.6875 -2.5t + t2 = 0 + -0.6875 Combine like terms: 0 + -0.6875 = -0.6875 -2.5t + t2 = -0.6875 The t term is -2.5t. Take half its coefficient (-1.25). Square it (1.5625) and add it to both sides. Add '1.5625' to each side of the equation. -2.5t + 1.5625 + t2 = -0.6875 + 1.5625 Reorder the terms: 1.5625 + -2.5t + t2 = -0.6875 + 1.5625 Combine like terms: -0.6875 + 1.5625 = 0.875 1.5625 + -2.5t + t2 = 0.875 Factor a perfect square on the left side: (t + -1.25)(t + -1.25) = 0.875 Calculate the square root of the right side: 0.935414347 Break this problem into two subproblems by setting (t + -1.25) equal to 0.935414347 and -0.935414347.Subproblem 1
t + -1.25 = 0.935414347 Simplifying t + -1.25 = 0.935414347 Reorder the terms: -1.25 + t = 0.935414347 Solving -1.25 + t = 0.935414347 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + t = 0.935414347 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + t = 0.935414347 + 1.25 t = 0.935414347 + 1.25 Combine like terms: 0.935414347 + 1.25 = 2.185414347 t = 2.185414347 Simplifying t = 2.185414347Subproblem 2
t + -1.25 = -0.935414347 Simplifying t + -1.25 = -0.935414347 Reorder the terms: -1.25 + t = -0.935414347 Solving -1.25 + t = -0.935414347 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + t = -0.935414347 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + t = -0.935414347 + 1.25 t = -0.935414347 + 1.25 Combine like terms: -0.935414347 + 1.25 = 0.314585653 t = 0.314585653 Simplifying t = 0.314585653Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.185414347, 0.314585653}
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