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Simplifying 3.5 = -5t2 + 20t + 1 Reorder the terms: 3.5 = 1 + 20t + -5t2 Solving 3.5 = 1 + 20t + -5t2 Solving for variable 't'. Combine like terms: 3.5 + -1 = 2.5 2.5 + -20t + 5t2 = 1 + 20t + -5t2 + -1 + -20t + 5t2 Reorder the terms: 2.5 + -20t + 5t2 = 1 + -1 + 20t + -20t + -5t2 + 5t2 Combine like terms: 1 + -1 = 0 2.5 + -20t + 5t2 = 0 + 20t + -20t + -5t2 + 5t2 2.5 + -20t + 5t2 = 20t + -20t + -5t2 + 5t2 Combine like terms: 20t + -20t = 0 2.5 + -20t + 5t2 = 0 + -5t2 + 5t2 2.5 + -20t + 5t2 = -5t2 + 5t2 Combine like terms: -5t2 + 5t2 = 0 2.5 + -20t + 5t2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. 0.5 + -4t + t2 = 0 Move the constant term to the right: Add '-0.5' to each side of the equation. 0.5 + -4t + -0.5 + t2 = 0 + -0.5 Reorder the terms: 0.5 + -0.5 + -4t + t2 = 0 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + -4t + t2 = 0 + -0.5 -4t + t2 = 0 + -0.5 Combine like terms: 0 + -0.5 = -0.5 -4t + t2 = -0.5 The t term is -4t. Take half its coefficient (-2). Square it (4) and add it to both sides. Add '4' to each side of the equation. -4t + 4 + t2 = -0.5 + 4 Reorder the terms: 4 + -4t + t2 = -0.5 + 4 Combine like terms: -0.5 + 4 = 3.5 4 + -4t + t2 = 3.5 Factor a perfect square on the left side: (t + -2)(t + -2) = 3.5 Calculate the square root of the right side: 1.870828693 Break this problem into two subproblems by setting (t + -2) equal to 1.870828693 and -1.870828693.Subproblem 1
t + -2 = 1.870828693 Simplifying t + -2 = 1.870828693 Reorder the terms: -2 + t = 1.870828693 Solving -2 + t = 1.870828693 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + t = 1.870828693 + 2 Combine like terms: -2 + 2 = 0 0 + t = 1.870828693 + 2 t = 1.870828693 + 2 Combine like terms: 1.870828693 + 2 = 3.870828693 t = 3.870828693 Simplifying t = 3.870828693Subproblem 2
t + -2 = -1.870828693 Simplifying t + -2 = -1.870828693 Reorder the terms: -2 + t = -1.870828693 Solving -2 + t = -1.870828693 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + t = -1.870828693 + 2 Combine like terms: -2 + 2 = 0 0 + t = -1.870828693 + 2 t = -1.870828693 + 2 Combine like terms: -1.870828693 + 2 = 0.129171307 t = 0.129171307 Simplifying t = 0.129171307Solution
The solution to the problem is based on the solutions from the subproblems. t = {3.870828693, 0.129171307}
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