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Simplifying 0 = -2t2 + 6t + 3.5 Reorder the terms: 0 = 3.5 + 6t + -2t2 Solving 0 = 3.5 + 6t + -2t2 Solving for variable 't'. Combine like terms: 0 + -3.5 = -3.5 -3.5 + -6t + 2t2 = 3.5 + 6t + -2t2 + -3.5 + -6t + 2t2 Reorder the terms: -3.5 + -6t + 2t2 = 3.5 + -3.5 + 6t + -6t + -2t2 + 2t2 Combine like terms: 3.5 + -3.5 = 0.0 -3.5 + -6t + 2t2 = 0.0 + 6t + -6t + -2t2 + 2t2 -3.5 + -6t + 2t2 = 6t + -6t + -2t2 + 2t2 Combine like terms: 6t + -6t = 0 -3.5 + -6t + 2t2 = 0 + -2t2 + 2t2 -3.5 + -6t + 2t2 = -2t2 + 2t2 Combine like terms: -2t2 + 2t2 = 0 -3.5 + -6t + 2t2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -1.75 + -3t + t2 = 0 Move the constant term to the right: Add '1.75' to each side of the equation. -1.75 + -3t + 1.75 + t2 = 0 + 1.75 Reorder the terms: -1.75 + 1.75 + -3t + t2 = 0 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + -3t + t2 = 0 + 1.75 -3t + t2 = 0 + 1.75 Combine like terms: 0 + 1.75 = 1.75 -3t + t2 = 1.75 The t term is -3t. Take half its coefficient (-1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. -3t + 2.25 + t2 = 1.75 + 2.25 Reorder the terms: 2.25 + -3t + t2 = 1.75 + 2.25 Combine like terms: 1.75 + 2.25 = 4 2.25 + -3t + t2 = 4 Factor a perfect square on the left side: (t + -1.5)(t + -1.5) = 4 Calculate the square root of the right side: 2 Break this problem into two subproblems by setting (t + -1.5) equal to 2 and -2.Subproblem 1
t + -1.5 = 2 Simplifying t + -1.5 = 2 Reorder the terms: -1.5 + t = 2 Solving -1.5 + t = 2 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + t = 2 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + t = 2 + 1.5 t = 2 + 1.5 Combine like terms: 2 + 1.5 = 3.5 t = 3.5 Simplifying t = 3.5Subproblem 2
t + -1.5 = -2 Simplifying t + -1.5 = -2 Reorder the terms: -1.5 + t = -2 Solving -1.5 + t = -2 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + t = -2 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + t = -2 + 1.5 t = -2 + 1.5 Combine like terms: -2 + 1.5 = -0.5 t = -0.5 Simplifying t = -0.5Solution
The solution to the problem is based on the solutions from the subproblems. t = {3.5, -0.5}
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