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Simplifying 2t(2t + 4) = 5t + 6 Reorder the terms: 2t(4 + 2t) = 5t + 6 (4 * 2t + 2t * 2t) = 5t + 6 (8t + 4t2) = 5t + 6 Reorder the terms: 8t + 4t2 = 6 + 5t Solving 8t + 4t2 = 6 + 5t Solving for variable 't'. Reorder the terms: -6 + 8t + -5t + 4t2 = 6 + 5t + -6 + -5t Combine like terms: 8t + -5t = 3t -6 + 3t + 4t2 = 6 + 5t + -6 + -5t Reorder the terms: -6 + 3t + 4t2 = 6 + -6 + 5t + -5t Combine like terms: 6 + -6 = 0 -6 + 3t + 4t2 = 0 + 5t + -5t -6 + 3t + 4t2 = 5t + -5t Combine like terms: 5t + -5t = 0 -6 + 3t + 4t2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -1.5 + 0.75t + t2 = 0 Move the constant term to the right: Add '1.5' to each side of the equation. -1.5 + 0.75t + 1.5 + t2 = 0 + 1.5 Reorder the terms: -1.5 + 1.5 + 0.75t + t2 = 0 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + 0.75t + t2 = 0 + 1.5 0.75t + t2 = 0 + 1.5 Combine like terms: 0 + 1.5 = 1.5 0.75t + t2 = 1.5 The t term is 0.75t. Take half its coefficient (0.375). Square it (0.140625) and add it to both sides. Add '0.140625' to each side of the equation. 0.75t + 0.140625 + t2 = 1.5 + 0.140625 Reorder the terms: 0.140625 + 0.75t + t2 = 1.5 + 0.140625 Combine like terms: 1.5 + 0.140625 = 1.640625 0.140625 + 0.75t + t2 = 1.640625 Factor a perfect square on the left side: (t + 0.375)(t + 0.375) = 1.640625 Calculate the square root of the right side: 1.280868846 Break this problem into two subproblems by setting (t + 0.375) equal to 1.280868846 and -1.280868846.Subproblem 1
t + 0.375 = 1.280868846 Simplifying t + 0.375 = 1.280868846 Reorder the terms: 0.375 + t = 1.280868846 Solving 0.375 + t = 1.280868846 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.375' to each side of the equation. 0.375 + -0.375 + t = 1.280868846 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + t = 1.280868846 + -0.375 t = 1.280868846 + -0.375 Combine like terms: 1.280868846 + -0.375 = 0.905868846 t = 0.905868846 Simplifying t = 0.905868846Subproblem 2
t + 0.375 = -1.280868846 Simplifying t + 0.375 = -1.280868846 Reorder the terms: 0.375 + t = -1.280868846 Solving 0.375 + t = -1.280868846 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.375' to each side of the equation. 0.375 + -0.375 + t = -1.280868846 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + t = -1.280868846 + -0.375 t = -1.280868846 + -0.375 Combine like terms: -1.280868846 + -0.375 = -1.655868846 t = -1.655868846 Simplifying t = -1.655868846Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.905868846, -1.655868846}
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