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Simplifying 5 = 3 + 9t + -5t2 Solving 5 = 3 + 9t + -5t2 Solving for variable 't'. Combine like terms: 5 + -3 = 2 2 + -9t + 5t2 = 3 + 9t + -5t2 + -3 + -9t + 5t2 Reorder the terms: 2 + -9t + 5t2 = 3 + -3 + 9t + -9t + -5t2 + 5t2 Combine like terms: 3 + -3 = 0 2 + -9t + 5t2 = 0 + 9t + -9t + -5t2 + 5t2 2 + -9t + 5t2 = 9t + -9t + -5t2 + 5t2 Combine like terms: 9t + -9t = 0 2 + -9t + 5t2 = 0 + -5t2 + 5t2 2 + -9t + 5t2 = -5t2 + 5t2 Combine like terms: -5t2 + 5t2 = 0 2 + -9t + 5t2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. 0.4 + -1.8t + t2 = 0 Move the constant term to the right: Add '-0.4' to each side of the equation. 0.4 + -1.8t + -0.4 + t2 = 0 + -0.4 Reorder the terms: 0.4 + -0.4 + -1.8t + t2 = 0 + -0.4 Combine like terms: 0.4 + -0.4 = 0.0 0.0 + -1.8t + t2 = 0 + -0.4 -1.8t + t2 = 0 + -0.4 Combine like terms: 0 + -0.4 = -0.4 -1.8t + t2 = -0.4 The t term is -1.8t. Take half its coefficient (-0.9). Square it (0.81) and add it to both sides. Add '0.81' to each side of the equation. -1.8t + 0.81 + t2 = -0.4 + 0.81 Reorder the terms: 0.81 + -1.8t + t2 = -0.4 + 0.81 Combine like terms: -0.4 + 0.81 = 0.41 0.81 + -1.8t + t2 = 0.41 Factor a perfect square on the left side: (t + -0.9)(t + -0.9) = 0.41 Calculate the square root of the right side: 0.640312424 Break this problem into two subproblems by setting (t + -0.9) equal to 0.640312424 and -0.640312424.Subproblem 1
t + -0.9 = 0.640312424 Simplifying t + -0.9 = 0.640312424 Reorder the terms: -0.9 + t = 0.640312424 Solving -0.9 + t = 0.640312424 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.9' to each side of the equation. -0.9 + 0.9 + t = 0.640312424 + 0.9 Combine like terms: -0.9 + 0.9 = 0.0 0.0 + t = 0.640312424 + 0.9 t = 0.640312424 + 0.9 Combine like terms: 0.640312424 + 0.9 = 1.540312424 t = 1.540312424 Simplifying t = 1.540312424Subproblem 2
t + -0.9 = -0.640312424 Simplifying t + -0.9 = -0.640312424 Reorder the terms: -0.9 + t = -0.640312424 Solving -0.9 + t = -0.640312424 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.9' to each side of the equation. -0.9 + 0.9 + t = -0.640312424 + 0.9 Combine like terms: -0.9 + 0.9 = 0.0 0.0 + t = -0.640312424 + 0.9 t = -0.640312424 + 0.9 Combine like terms: -0.640312424 + 0.9 = 0.259687576 t = 0.259687576 Simplifying t = 0.259687576Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.540312424, 0.259687576}
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