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Simplifying 40(t) = -16t2 + 25 + 6 Reorder the terms: 40t = 25 + 6 + -16t2 Combine like terms: 25 + 6 = 31 40t = 31 + -16t2 Solving 40t = 31 + -16t2 Solving for variable 't'. Reorder the terms: -31 + 40t + 16t2 = 31 + -16t2 + -31 + 16t2 Reorder the terms: -31 + 40t + 16t2 = 31 + -31 + -16t2 + 16t2 Combine like terms: 31 + -31 = 0 -31 + 40t + 16t2 = 0 + -16t2 + 16t2 -31 + 40t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -31 + 40t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -1.9375 + 2.5t + t2 = 0 Move the constant term to the right: Add '1.9375' to each side of the equation. -1.9375 + 2.5t + 1.9375 + t2 = 0 + 1.9375 Reorder the terms: -1.9375 + 1.9375 + 2.5t + t2 = 0 + 1.9375 Combine like terms: -1.9375 + 1.9375 = 0.0000 0.0000 + 2.5t + t2 = 0 + 1.9375 2.5t + t2 = 0 + 1.9375 Combine like terms: 0 + 1.9375 = 1.9375 2.5t + t2 = 1.9375 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 = 1.9375 + 1.5625 Reorder the terms: 1.5625 + 2.5t + t2 = 1.9375 + 1.5625 Combine like terms: 1.9375 + 1.5625 = 3.5 1.5625 + 2.5t + t2 = 3.5 Factor a perfect square on the left side: (t + 1.25)(t + 1.25) = 3.5 Calculate the square root of the right side: 1.870828693 Break this problem into two subproblems by setting (t + 1.25) equal to 1.870828693 and -1.870828693.Subproblem 1
t + 1.25 = 1.870828693 Simplifying t + 1.25 = 1.870828693 Reorder the terms: 1.25 + t = 1.870828693 Solving 1.25 + t = 1.870828693 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 = 1.870828693 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + t = 1.870828693 + -1.25 t = 1.870828693 + -1.25 Combine like terms: 1.870828693 + -1.25 = 0.620828693 t = 0.620828693 Simplifying t = 0.620828693Subproblem 2
t + 1.25 = -1.870828693 Simplifying t + 1.25 = -1.870828693 Reorder the terms: 1.25 + t = -1.870828693 Solving 1.25 + t = -1.870828693 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 = -1.870828693 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + t = -1.870828693 + -1.25 t = -1.870828693 + -1.25 Combine like terms: -1.870828693 + -1.25 = -3.120828693 t = -3.120828693 Simplifying t = -3.120828693Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.620828693, -3.120828693}
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