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Simplifying 5(t2 + -3t + 2) = (5t + -2) Reorder the terms: 5(2 + -3t + t2) = (5t + -2) (2 * 5 + -3t * 5 + t2 * 5) = (5t + -2) (10 + -15t + 5t2) = (5t + -2) Reorder the terms: 10 + -15t + 5t2 = (-2 + 5t) Remove parenthesis around (-2 + 5t) 10 + -15t + 5t2 = -2 + 5t Solving 10 + -15t + 5t2 = -2 + 5t Solving for variable 't'. Reorder the terms: 10 + 2 + -15t + -5t + 5t2 = -2 + 5t + 2 + -5t Combine like terms: 10 + 2 = 12 12 + -15t + -5t + 5t2 = -2 + 5t + 2 + -5t Combine like terms: -15t + -5t = -20t 12 + -20t + 5t2 = -2 + 5t + 2 + -5t Reorder the terms: 12 + -20t + 5t2 = -2 + 2 + 5t + -5t Combine like terms: -2 + 2 = 0 12 + -20t + 5t2 = 0 + 5t + -5t 12 + -20t + 5t2 = 5t + -5t Combine like terms: 5t + -5t = 0 12 + -20t + 5t2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. 2.4 + -4t + t2 = 0 Move the constant term to the right: Add '-2.4' to each side of the equation. 2.4 + -4t + -2.4 + t2 = 0 + -2.4 Reorder the terms: 2.4 + -2.4 + -4t + t2 = 0 + -2.4 Combine like terms: 2.4 + -2.4 = 0.0 0.0 + -4t + t2 = 0 + -2.4 -4t + t2 = 0 + -2.4 Combine like terms: 0 + -2.4 = -2.4 -4t + t2 = -2.4 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 = -2.4 + 4 Reorder the terms: 4 + -4t + t2 = -2.4 + 4 Combine like terms: -2.4 + 4 = 1.6 4 + -4t + t2 = 1.6 Factor a perfect square on the left side: (t + -2)(t + -2) = 1.6 Calculate the square root of the right side: 1.264911064 Break this problem into two subproblems by setting (t + -2) equal to 1.264911064 and -1.264911064.Subproblem 1
t + -2 = 1.264911064 Simplifying t + -2 = 1.264911064 Reorder the terms: -2 + t = 1.264911064 Solving -2 + t = 1.264911064 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 = 1.264911064 + 2 Combine like terms: -2 + 2 = 0 0 + t = 1.264911064 + 2 t = 1.264911064 + 2 Combine like terms: 1.264911064 + 2 = 3.264911064 t = 3.264911064 Simplifying t = 3.264911064Subproblem 2
t + -2 = -1.264911064 Simplifying t + -2 = -1.264911064 Reorder the terms: -2 + t = -1.264911064 Solving -2 + t = -1.264911064 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 = -1.264911064 + 2 Combine like terms: -2 + 2 = 0 0 + t = -1.264911064 + 2 t = -1.264911064 + 2 Combine like terms: -1.264911064 + 2 = 0.735088936 t = 0.735088936 Simplifying t = 0.735088936Solution
The solution to the problem is based on the solutions from the subproblems. t = {3.264911064, 0.735088936}
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