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Simplifying t(9t + 11) = 6.5 Reorder the terms: t(11 + 9t) = 6.5 (11 * t + 9t * t) = 6.5 (11t + 9t2) = 6.5 Solving 11t + 9t2 = 6.5 Solving for variable 't'. Reorder the terms: -6.5 + 11t + 9t2 = 6.5 + -6.5 Combine like terms: 6.5 + -6.5 = 0.0 -6.5 + 11t + 9t2 = 0.0 Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -0.7222222222 + 1.222222222t + t2 = 0.0 Move the constant term to the right: Add '0.7222222222' to each side of the equation. -0.7222222222 + 1.222222222t + 0.7222222222 + t2 = 0.0 + 0.7222222222 Reorder the terms: -0.7222222222 + 0.7222222222 + 1.222222222t + t2 = 0.0 + 0.7222222222 Combine like terms: -0.7222222222 + 0.7222222222 = 0.0000000000 0.0000000000 + 1.222222222t + t2 = 0.0 + 0.7222222222 1.222222222t + t2 = 0.0 + 0.7222222222 Combine like terms: 0.0 + 0.7222222222 = 0.7222222222 1.222222222t + t2 = 0.7222222222 The t term is 1.222222222t. Take half its coefficient (0.611111111). Square it (0.3734567900) and add it to both sides. Add '0.3734567900' to each side of the equation. 1.222222222t + 0.3734567900 + t2 = 0.7222222222 + 0.3734567900 Reorder the terms: 0.3734567900 + 1.222222222t + t2 = 0.7222222222 + 0.3734567900 Combine like terms: 0.7222222222 + 0.3734567900 = 1.0956790122 0.3734567900 + 1.222222222t + t2 = 1.0956790122 Factor a perfect square on the left side: (t + 0.611111111)(t + 0.611111111) = 1.0956790122 Calculate the square root of the right side: 1.046746871 Break this problem into two subproblems by setting (t + 0.611111111) equal to 1.046746871 and -1.046746871.Subproblem 1
t + 0.611111111 = 1.046746871 Simplifying t + 0.611111111 = 1.046746871 Reorder the terms: 0.611111111 + t = 1.046746871 Solving 0.611111111 + t = 1.046746871 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.611111111' to each side of the equation. 0.611111111 + -0.611111111 + t = 1.046746871 + -0.611111111 Combine like terms: 0.611111111 + -0.611111111 = 0.000000000 0.000000000 + t = 1.046746871 + -0.611111111 t = 1.046746871 + -0.611111111 Combine like terms: 1.046746871 + -0.611111111 = 0.43563576 t = 0.43563576 Simplifying t = 0.43563576Subproblem 2
t + 0.611111111 = -1.046746871 Simplifying t + 0.611111111 = -1.046746871 Reorder the terms: 0.611111111 + t = -1.046746871 Solving 0.611111111 + t = -1.046746871 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.611111111' to each side of the equation. 0.611111111 + -0.611111111 + t = -1.046746871 + -0.611111111 Combine like terms: 0.611111111 + -0.611111111 = 0.000000000 0.000000000 + t = -1.046746871 + -0.611111111 t = -1.046746871 + -0.611111111 Combine like terms: -1.046746871 + -0.611111111 = -1.657857982 t = -1.657857982 Simplifying t = -1.657857982Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.43563576, -1.657857982}
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