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Simplifying 0 = -40t2 + 30t + 6 Reorder the terms: 0 = 6 + 30t + -40t2 Solving 0 = 6 + 30t + -40t2 Solving for variable 't'. Combine like terms: 0 + -6 = -6 -6 + -30t + 40t2 = 6 + 30t + -40t2 + -6 + -30t + 40t2 Reorder the terms: -6 + -30t + 40t2 = 6 + -6 + 30t + -30t + -40t2 + 40t2 Combine like terms: 6 + -6 = 0 -6 + -30t + 40t2 = 0 + 30t + -30t + -40t2 + 40t2 -6 + -30t + 40t2 = 30t + -30t + -40t2 + 40t2 Combine like terms: 30t + -30t = 0 -6 + -30t + 40t2 = 0 + -40t2 + 40t2 -6 + -30t + 40t2 = -40t2 + 40t2 Combine like terms: -40t2 + 40t2 = 0 -6 + -30t + 40t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-3 + -15t + 20t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-3 + -15t + 20t2)' equal to zero and attempt to solve: Simplifying -3 + -15t + 20t2 = 0 Solving -3 + -15t + 20t2 = 0 Begin completing the square. Divide all terms by 20 the coefficient of the squared term: Divide each side by '20'. -0.15 + -0.75t + t2 = 0.0 Move the constant term to the right: Add '0.15' to each side of the equation. -0.15 + -0.75t + 0.15 + t2 = 0.0 + 0.15 Reorder the terms: -0.15 + 0.15 + -0.75t + t2 = 0.0 + 0.15 Combine like terms: -0.15 + 0.15 = 0.00 0.00 + -0.75t + t2 = 0.0 + 0.15 -0.75t + t2 = 0.0 + 0.15 Combine like terms: 0.0 + 0.15 = 0.15 -0.75t + t2 = 0.15 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 = 0.15 + 0.140625 Reorder the terms: 0.140625 + -0.75t + t2 = 0.15 + 0.140625 Combine like terms: 0.15 + 0.140625 = 0.290625 0.140625 + -0.75t + t2 = 0.290625 Factor a perfect square on the left side: (t + -0.375)(t + -0.375) = 0.290625 Calculate the square root of the right side: 0.539096466 Break this problem into two subproblems by setting (t + -0.375) equal to 0.539096466 and -0.539096466.Subproblem 1
t + -0.375 = 0.539096466 Simplifying t + -0.375 = 0.539096466 Reorder the terms: -0.375 + t = 0.539096466 Solving -0.375 + t = 0.539096466 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 = 0.539096466 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + t = 0.539096466 + 0.375 t = 0.539096466 + 0.375 Combine like terms: 0.539096466 + 0.375 = 0.914096466 t = 0.914096466 Simplifying t = 0.914096466Subproblem 2
t + -0.375 = -0.539096466 Simplifying t + -0.375 = -0.539096466 Reorder the terms: -0.375 + t = -0.539096466 Solving -0.375 + t = -0.539096466 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 = -0.539096466 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + t = -0.539096466 + 0.375 t = -0.539096466 + 0.375 Combine like terms: -0.539096466 + 0.375 = -0.164096466 t = -0.164096466 Simplifying t = -0.164096466Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.914096466, -0.164096466}Solution
t = {0.914096466, -0.164096466}
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