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Simplifying 3t = t2 + 2t + -7 Reorder the terms: 3t = -7 + 2t + t2 Solving 3t = -7 + 2t + t2 Solving for variable 't'. Reorder the terms: 7 + 3t + -2t + -1t2 = -7 + 2t + t2 + 7 + -2t + -1t2 Combine like terms: 3t + -2t = 1t 7 + 1t + -1t2 = -7 + 2t + t2 + 7 + -2t + -1t2 Reorder the terms: 7 + 1t + -1t2 = -7 + 7 + 2t + -2t + t2 + -1t2 Combine like terms: -7 + 7 = 0 7 + 1t + -1t2 = 0 + 2t + -2t + t2 + -1t2 7 + 1t + -1t2 = 2t + -2t + t2 + -1t2 Combine like terms: 2t + -2t = 0 7 + 1t + -1t2 = 0 + t2 + -1t2 7 + 1t + -1t2 = t2 + -1t2 Combine like terms: t2 + -1t2 = 0 7 + 1t + -1t2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -7 + -1t + t2 = 0 Move the constant term to the right: Add '7' to each side of the equation. -7 + -1t + 7 + t2 = 0 + 7 Reorder the terms: -7 + 7 + -1t + t2 = 0 + 7 Combine like terms: -7 + 7 = 0 0 + -1t + t2 = 0 + 7 -1t + t2 = 0 + 7 Combine like terms: 0 + 7 = 7 -1t + t2 = 7 The t term is -1t. Take half its coefficient (-0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -1t + 0.25 + t2 = 7 + 0.25 Reorder the terms: 0.25 + -1t + t2 = 7 + 0.25 Combine like terms: 7 + 0.25 = 7.25 0.25 + -1t + t2 = 7.25 Factor a perfect square on the left side: (t + -0.5)(t + -0.5) = 7.25 Calculate the square root of the right side: 2.692582404 Break this problem into two subproblems by setting (t + -0.5) equal to 2.692582404 and -2.692582404.Subproblem 1
t + -0.5 = 2.692582404 Simplifying t + -0.5 = 2.692582404 Reorder the terms: -0.5 + t = 2.692582404 Solving -0.5 + t = 2.692582404 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + t = 2.692582404 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + t = 2.692582404 + 0.5 t = 2.692582404 + 0.5 Combine like terms: 2.692582404 + 0.5 = 3.192582404 t = 3.192582404 Simplifying t = 3.192582404Subproblem 2
t + -0.5 = -2.692582404 Simplifying t + -0.5 = -2.692582404 Reorder the terms: -0.5 + t = -2.692582404 Solving -0.5 + t = -2.692582404 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + t = -2.692582404 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + t = -2.692582404 + 0.5 t = -2.692582404 + 0.5 Combine like terms: -2.692582404 + 0.5 = -2.192582404 t = -2.192582404 Simplifying t = -2.192582404Solution
The solution to the problem is based on the solutions from the subproblems. t = {3.192582404, -2.192582404}
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