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Simplifying (6t + 9)(6t + 9) = 14 Reorder the terms: (9 + 6t)(6t + 9) = 14 Reorder the terms: (9 + 6t)(9 + 6t) = 14 Multiply (9 + 6t) * (9 + 6t) (9(9 + 6t) + 6t * (9 + 6t)) = 14 ((9 * 9 + 6t * 9) + 6t * (9 + 6t)) = 14 ((81 + 54t) + 6t * (9 + 6t)) = 14 (81 + 54t + (9 * 6t + 6t * 6t)) = 14 (81 + 54t + (54t + 36t2)) = 14 Combine like terms: 54t + 54t = 108t (81 + 108t + 36t2) = 14 Solving 81 + 108t + 36t2 = 14 Solving for variable 't'. Reorder the terms: 81 + -14 + 108t + 36t2 = 14 + -14 Combine like terms: 81 + -14 = 67 67 + 108t + 36t2 = 14 + -14 Combine like terms: 14 + -14 = 0 67 + 108t + 36t2 = 0 Begin completing the square. Divide all terms by 36 the coefficient of the squared term: Divide each side by '36'. 1.861111111 + 3t + t2 = 0 Move the constant term to the right: Add '-1.861111111' to each side of the equation. 1.861111111 + 3t + -1.861111111 + t2 = 0 + -1.861111111 Reorder the terms: 1.861111111 + -1.861111111 + 3t + t2 = 0 + -1.861111111 Combine like terms: 1.861111111 + -1.861111111 = 0.000000000 0.000000000 + 3t + t2 = 0 + -1.861111111 3t + t2 = 0 + -1.861111111 Combine like terms: 0 + -1.861111111 = -1.861111111 3t + t2 = -1.861111111 The t term is 3t. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3t + 2.25 + t2 = -1.861111111 + 2.25 Reorder the terms: 2.25 + 3t + t2 = -1.861111111 + 2.25 Combine like terms: -1.861111111 + 2.25 = 0.388888889 2.25 + 3t + t2 = 0.388888889 Factor a perfect square on the left side: (t + 1.5)(t + 1.5) = 0.388888889 Calculate the square root of the right side: 0.623609565 Break this problem into two subproblems by setting (t + 1.5) equal to 0.623609565 and -0.623609565.Subproblem 1
t + 1.5 = 0.623609565 Simplifying t + 1.5 = 0.623609565 Reorder the terms: 1.5 + t = 0.623609565 Solving 1.5 + t = 0.623609565 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + t = 0.623609565 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + t = 0.623609565 + -1.5 t = 0.623609565 + -1.5 Combine like terms: 0.623609565 + -1.5 = -0.876390435 t = -0.876390435 Simplifying t = -0.876390435Subproblem 2
t + 1.5 = -0.623609565 Simplifying t + 1.5 = -0.623609565 Reorder the terms: 1.5 + t = -0.623609565 Solving 1.5 + t = -0.623609565 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + t = -0.623609565 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + t = -0.623609565 + -1.5 t = -0.623609565 + -1.5 Combine like terms: -0.623609565 + -1.5 = -2.123609565 t = -2.123609565 Simplifying t = -2.123609565Solution
The solution to the problem is based on the solutions from the subproblems. t = {-0.876390435, -2.123609565}
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