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Simplifying 2t(t + -3) + 3 = 0 Reorder the terms: 2t(-3 + t) + 3 = 0 (-3 * 2t + t * 2t) + 3 = 0 (-6t + 2t2) + 3 = 0 Reorder the terms: 3 + -6t + 2t2 = 0 Solving 3 + -6t + 2t2 = 0 Solving for variable 't'. Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 1.5 + -3t + t2 = 0 Move the constant term to the right: Add '-1.5' to each side of the equation. 1.5 + -3t + -1.5 + t2 = 0 + -1.5 Reorder the terms: 1.5 + -1.5 + -3t + t2 = 0 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + -3t + t2 = 0 + -1.5 -3t + t2 = 0 + -1.5 Combine like terms: 0 + -1.5 = -1.5 -3t + t2 = -1.5 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 = -1.5 + 2.25 Reorder the terms: 2.25 + -3t + t2 = -1.5 + 2.25 Combine like terms: -1.5 + 2.25 = 0.75 2.25 + -3t + t2 = 0.75 Factor a perfect square on the left side: (t + -1.5)(t + -1.5) = 0.75 Calculate the square root of the right side: 0.866025404 Break this problem into two subproblems by setting (t + -1.5) equal to 0.866025404 and -0.866025404.Subproblem 1
t + -1.5 = 0.866025404 Simplifying t + -1.5 = 0.866025404 Reorder the terms: -1.5 + t = 0.866025404 Solving -1.5 + t = 0.866025404 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 = 0.866025404 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + t = 0.866025404 + 1.5 t = 0.866025404 + 1.5 Combine like terms: 0.866025404 + 1.5 = 2.366025404 t = 2.366025404 Simplifying t = 2.366025404Subproblem 2
t + -1.5 = -0.866025404 Simplifying t + -1.5 = -0.866025404 Reorder the terms: -1.5 + t = -0.866025404 Solving -1.5 + t = -0.866025404 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 = -0.866025404 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + t = -0.866025404 + 1.5 t = -0.866025404 + 1.5 Combine like terms: -0.866025404 + 1.5 = 0.633974596 t = 0.633974596 Simplifying t = 0.633974596Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.366025404, 0.633974596}
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