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Simplifying t2 = 3t + 5 Reorder the terms: t2 = 5 + 3t Solving t2 = 5 + 3t Solving for variable 't'. Reorder the terms: -5 + -3t + t2 = 5 + 3t + -5 + -3t Reorder the terms: -5 + -3t + t2 = 5 + -5 + 3t + -3t Combine like terms: 5 + -5 = 0 -5 + -3t + t2 = 0 + 3t + -3t -5 + -3t + t2 = 3t + -3t Combine like terms: 3t + -3t = 0 -5 + -3t + t2 = 0 Begin completing the square. Move the constant term to the right: Add '5' to each side of the equation. -5 + -3t + 5 + t2 = 0 + 5 Reorder the terms: -5 + 5 + -3t + t2 = 0 + 5 Combine like terms: -5 + 5 = 0 0 + -3t + t2 = 0 + 5 -3t + t2 = 0 + 5 Combine like terms: 0 + 5 = 5 -3t + t2 = 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 = 5 + 2.25 Reorder the terms: 2.25 + -3t + t2 = 5 + 2.25 Combine like terms: 5 + 2.25 = 7.25 2.25 + -3t + t2 = 7.25 Factor a perfect square on the left side: (t + -1.5)(t + -1.5) = 7.25 Calculate the square root of the right side: 2.692582404 Break this problem into two subproblems by setting (t + -1.5) equal to 2.692582404 and -2.692582404.Subproblem 1
t + -1.5 = 2.692582404 Simplifying t + -1.5 = 2.692582404 Reorder the terms: -1.5 + t = 2.692582404 Solving -1.5 + t = 2.692582404 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 = 2.692582404 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + t = 2.692582404 + 1.5 t = 2.692582404 + 1.5 Combine like terms: 2.692582404 + 1.5 = 4.192582404 t = 4.192582404 Simplifying t = 4.192582404Subproblem 2
t + -1.5 = -2.692582404 Simplifying t + -1.5 = -2.692582404 Reorder the terms: -1.5 + t = -2.692582404 Solving -1.5 + t = -2.692582404 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 = -2.692582404 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + t = -2.692582404 + 1.5 t = -2.692582404 + 1.5 Combine like terms: -2.692582404 + 1.5 = -1.192582404 t = -1.192582404 Simplifying t = -1.192582404Solution
The solution to the problem is based on the solutions from the subproblems. t = {4.192582404, -1.192582404}
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