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Simplifying -16t2 + 104t + -167 = 0 Reorder the terms: -167 + 104t + -16t2 = 0 Solving -167 + 104t + -16t2 = 0 Solving for variable 't'. Begin completing the square. Divide all terms by -16 the coefficient of the squared term: Divide each side by '-16'. 10.4375 + -6.5t + t2 = 0 Move the constant term to the right: Add '-10.4375' to each side of the equation. 10.4375 + -6.5t + -10.4375 + t2 = 0 + -10.4375 Reorder the terms: 10.4375 + -10.4375 + -6.5t + t2 = 0 + -10.4375 Combine like terms: 10.4375 + -10.4375 = 0.0000 0.0000 + -6.5t + t2 = 0 + -10.4375 -6.5t + t2 = 0 + -10.4375 Combine like terms: 0 + -10.4375 = -10.4375 -6.5t + t2 = -10.4375 The t term is -6.5t. Take half its coefficient (-3.25). Square it (10.5625) and add it to both sides. Add '10.5625' to each side of the equation. -6.5t + 10.5625 + t2 = -10.4375 + 10.5625 Reorder the terms: 10.5625 + -6.5t + t2 = -10.4375 + 10.5625 Combine like terms: -10.4375 + 10.5625 = 0.125 10.5625 + -6.5t + t2 = 0.125 Factor a perfect square on the left side: (t + -3.25)(t + -3.25) = 0.125 Calculate the square root of the right side: 0.353553391 Break this problem into two subproblems by setting (t + -3.25) equal to 0.353553391 and -0.353553391.Subproblem 1
t + -3.25 = 0.353553391 Simplifying t + -3.25 = 0.353553391 Reorder the terms: -3.25 + t = 0.353553391 Solving -3.25 + t = 0.353553391 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3.25' to each side of the equation. -3.25 + 3.25 + t = 0.353553391 + 3.25 Combine like terms: -3.25 + 3.25 = 0.00 0.00 + t = 0.353553391 + 3.25 t = 0.353553391 + 3.25 Combine like terms: 0.353553391 + 3.25 = 3.603553391 t = 3.603553391 Simplifying t = 3.603553391Subproblem 2
t + -3.25 = -0.353553391 Simplifying t + -3.25 = -0.353553391 Reorder the terms: -3.25 + t = -0.353553391 Solving -3.25 + t = -0.353553391 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3.25' to each side of the equation. -3.25 + 3.25 + t = -0.353553391 + 3.25 Combine like terms: -3.25 + 3.25 = 0.00 0.00 + t = -0.353553391 + 3.25 t = -0.353553391 + 3.25 Combine like terms: -0.353553391 + 3.25 = 2.896446609 t = 2.896446609 Simplifying t = 2.896446609Solution
The solution to the problem is based on the solutions from the subproblems. t = {3.603553391, 2.896446609}
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