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Simplifying 2t = 2 + 3t + -5t2 Solving 2t = 2 + 3t + -5t2 Solving for variable 't'. Reorder the terms: -2 + 2t + -3t + 5t2 = 2 + 3t + -5t2 + -2 + -3t + 5t2 Combine like terms: 2t + -3t = -1t -2 + -1t + 5t2 = 2 + 3t + -5t2 + -2 + -3t + 5t2 Reorder the terms: -2 + -1t + 5t2 = 2 + -2 + 3t + -3t + -5t2 + 5t2 Combine like terms: 2 + -2 = 0 -2 + -1t + 5t2 = 0 + 3t + -3t + -5t2 + 5t2 -2 + -1t + 5t2 = 3t + -3t + -5t2 + 5t2 Combine like terms: 3t + -3t = 0 -2 + -1t + 5t2 = 0 + -5t2 + 5t2 -2 + -1t + 5t2 = -5t2 + 5t2 Combine like terms: -5t2 + 5t2 = 0 -2 + -1t + 5t2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -0.4 + -0.2t + t2 = 0 Move the constant term to the right: Add '0.4' to each side of the equation. -0.4 + -0.2t + 0.4 + t2 = 0 + 0.4 Reorder the terms: -0.4 + 0.4 + -0.2t + t2 = 0 + 0.4 Combine like terms: -0.4 + 0.4 = 0.0 0.0 + -0.2t + t2 = 0 + 0.4 -0.2t + t2 = 0 + 0.4 Combine like terms: 0 + 0.4 = 0.4 -0.2t + t2 = 0.4 The t term is -0.2t. Take half its coefficient (-0.1). Square it (0.01) and add it to both sides. Add '0.01' to each side of the equation. -0.2t + 0.01 + t2 = 0.4 + 0.01 Reorder the terms: 0.01 + -0.2t + t2 = 0.4 + 0.01 Combine like terms: 0.4 + 0.01 = 0.41 0.01 + -0.2t + t2 = 0.41 Factor a perfect square on the left side: (t + -0.1)(t + -0.1) = 0.41 Calculate the square root of the right side: 0.640312424 Break this problem into two subproblems by setting (t + -0.1) equal to 0.640312424 and -0.640312424.Subproblem 1
t + -0.1 = 0.640312424 Simplifying t + -0.1 = 0.640312424 Reorder the terms: -0.1 + t = 0.640312424 Solving -0.1 + t = 0.640312424 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.1' to each side of the equation. -0.1 + 0.1 + t = 0.640312424 + 0.1 Combine like terms: -0.1 + 0.1 = 0.0 0.0 + t = 0.640312424 + 0.1 t = 0.640312424 + 0.1 Combine like terms: 0.640312424 + 0.1 = 0.740312424 t = 0.740312424 Simplifying t = 0.740312424Subproblem 2
t + -0.1 = -0.640312424 Simplifying t + -0.1 = -0.640312424 Reorder the terms: -0.1 + t = -0.640312424 Solving -0.1 + t = -0.640312424 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.1' to each side of the equation. -0.1 + 0.1 + t = -0.640312424 + 0.1 Combine like terms: -0.1 + 0.1 = 0.0 0.0 + t = -0.640312424 + 0.1 t = -0.640312424 + 0.1 Combine like terms: -0.640312424 + 0.1 = -0.540312424 t = -0.540312424 Simplifying t = -0.540312424Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.740312424, -0.540312424}
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