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Simplifying 0.5(8t2 + 14t2 + 0.5t) = 24 Reorder the terms: 0.5(0.5t + 8t2 + 14t2) = 24 Combine like terms: 8t2 + 14t2 = 22t2 0.5(0.5t + 22t2) = 24 (0.5t * 0.5 + 22t2 * 0.5) = 24 (0.25t + 11t2) = 24 Solving 0.25t + 11t2 = 24 Solving for variable 't'. Reorder the terms: -24 + 0.25t + 11t2 = 24 + -24 Combine like terms: 24 + -24 = 0 -24 + 0.25t + 11t2 = 0 Begin completing the square. Divide all terms by 11 the coefficient of the squared term: Divide each side by '11'. -2.181818182 + 0.02272727273t + t2 = 0 Move the constant term to the right: Add '2.181818182' to each side of the equation. -2.181818182 + 0.02272727273t + 2.181818182 + t2 = 0 + 2.181818182 Reorder the terms: -2.181818182 + 2.181818182 + 0.02272727273t + t2 = 0 + 2.181818182 Combine like terms: -2.181818182 + 2.181818182 = 0.000000000 0.000000000 + 0.02272727273t + t2 = 0 + 2.181818182 0.02272727273t + t2 = 0 + 2.181818182 Combine like terms: 0 + 2.181818182 = 2.181818182 0.02272727273t + t2 = 2.181818182 The t term is 0.02272727273t. Take half its coefficient (0.01136363637). Square it (0.0001291322316) and add it to both sides. Add '0.0001291322316' to each side of the equation. 0.02272727273t + 0.0001291322316 + t2 = 2.181818182 + 0.0001291322316 Reorder the terms: 0.0001291322316 + 0.02272727273t + t2 = 2.181818182 + 0.0001291322316 Combine like terms: 2.181818182 + 0.0001291322316 = 2.1819473142316 0.0001291322316 + 0.02272727273t + t2 = 2.1819473142316 Factor a perfect square on the left side: (t + 0.01136363637)(t + 0.01136363637) = 2.1819473142316 Calculate the square root of the right side: 1.477141603 Break this problem into two subproblems by setting (t + 0.01136363637) equal to 1.477141603 and -1.477141603.Subproblem 1
t + 0.01136363637 = 1.477141603 Simplifying t + 0.01136363637 = 1.477141603 Reorder the terms: 0.01136363637 + t = 1.477141603 Solving 0.01136363637 + t = 1.477141603 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.01136363637' to each side of the equation. 0.01136363637 + -0.01136363637 + t = 1.477141603 + -0.01136363637 Combine like terms: 0.01136363637 + -0.01136363637 = 0.00000000000 0.00000000000 + t = 1.477141603 + -0.01136363637 t = 1.477141603 + -0.01136363637 Combine like terms: 1.477141603 + -0.01136363637 = 1.46577796663 t = 1.46577796663 Simplifying t = 1.46577796663Subproblem 2
t + 0.01136363637 = -1.477141603 Simplifying t + 0.01136363637 = -1.477141603 Reorder the terms: 0.01136363637 + t = -1.477141603 Solving 0.01136363637 + t = -1.477141603 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.01136363637' to each side of the equation. 0.01136363637 + -0.01136363637 + t = -1.477141603 + -0.01136363637 Combine like terms: 0.01136363637 + -0.01136363637 = 0.00000000000 0.00000000000 + t = -1.477141603 + -0.01136363637 t = -1.477141603 + -0.01136363637 Combine like terms: -1.477141603 + -0.01136363637 = -1.48850523937 t = -1.48850523937 Simplifying t = -1.48850523937Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.46577796663, -1.48850523937}
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