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Simplifying 6t2 + 9t = 8 Reorder the terms: 9t + 6t2 = 8 Solving 9t + 6t2 = 8 Solving for variable 't'. Reorder the terms: -8 + 9t + 6t2 = 8 + -8 Combine like terms: 8 + -8 = 0 -8 + 9t + 6t2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -1.333333333 + 1.5t + t2 = 0 Move the constant term to the right: Add '1.333333333' to each side of the equation. -1.333333333 + 1.5t + 1.333333333 + t2 = 0 + 1.333333333 Reorder the terms: -1.333333333 + 1.333333333 + 1.5t + t2 = 0 + 1.333333333 Combine like terms: -1.333333333 + 1.333333333 = 0.000000000 0.000000000 + 1.5t + t2 = 0 + 1.333333333 1.5t + t2 = 0 + 1.333333333 Combine like terms: 0 + 1.333333333 = 1.333333333 1.5t + t2 = 1.333333333 The t term is 1.5t. Take half its coefficient (0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. 1.5t + 0.5625 + t2 = 1.333333333 + 0.5625 Reorder the terms: 0.5625 + 1.5t + t2 = 1.333333333 + 0.5625 Combine like terms: 1.333333333 + 0.5625 = 1.895833333 0.5625 + 1.5t + t2 = 1.895833333 Factor a perfect square on the left side: (t + 0.75)(t + 0.75) = 1.895833333 Calculate the square root of the right side: 1.376892637 Break this problem into two subproblems by setting (t + 0.75) equal to 1.376892637 and -1.376892637.Subproblem 1
t + 0.75 = 1.376892637 Simplifying t + 0.75 = 1.376892637 Reorder the terms: 0.75 + t = 1.376892637 Solving 0.75 + t = 1.376892637 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + t = 1.376892637 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + t = 1.376892637 + -0.75 t = 1.376892637 + -0.75 Combine like terms: 1.376892637 + -0.75 = 0.626892637 t = 0.626892637 Simplifying t = 0.626892637Subproblem 2
t + 0.75 = -1.376892637 Simplifying t + 0.75 = -1.376892637 Reorder the terms: 0.75 + t = -1.376892637 Solving 0.75 + t = -1.376892637 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + t = -1.376892637 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + t = -1.376892637 + -0.75 t = -1.376892637 + -0.75 Combine like terms: -1.376892637 + -0.75 = -2.126892637 t = -2.126892637 Simplifying t = -2.126892637Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.626892637, -2.126892637}
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