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Simplifying 3(2t + -1) * 2t = 6 + -1t Reorder the terms: 3(-1 + 2t) * 2t = 6 + -1t Reorder the terms for easier multiplication: 3 * 2t(-1 + 2t) = 6 + -1t Multiply 3 * 2 6t(-1 + 2t) = 6 + -1t (-1 * 6t + 2t * 6t) = 6 + -1t (-6t + 12t2) = 6 + -1t Solving -6t + 12t2 = 6 + -1t Solving for variable 't'. Reorder the terms: -6 + -6t + t + 12t2 = 6 + -1t + -6 + t Combine like terms: -6t + t = -5t -6 + -5t + 12t2 = 6 + -1t + -6 + t Reorder the terms: -6 + -5t + 12t2 = 6 + -6 + -1t + t Combine like terms: 6 + -6 = 0 -6 + -5t + 12t2 = 0 + -1t + t -6 + -5t + 12t2 = -1t + t Combine like terms: -1t + t = 0 -6 + -5t + 12t2 = 0 Begin completing the square. Divide all terms by 12 the coefficient of the squared term: Divide each side by '12'. -0.5 + -0.4166666667t + t2 = 0 Move the constant term to the right: Add '0.5' to each side of the equation. -0.5 + -0.4166666667t + 0.5 + t2 = 0 + 0.5 Reorder the terms: -0.5 + 0.5 + -0.4166666667t + t2 = 0 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + -0.4166666667t + t2 = 0 + 0.5 -0.4166666667t + t2 = 0 + 0.5 Combine like terms: 0 + 0.5 = 0.5 -0.4166666667t + t2 = 0.5 The t term is -0.4166666667t. Take half its coefficient (-0.2083333334). Square it (0.04340277781) and add it to both sides. Add '0.04340277781' to each side of the equation. -0.4166666667t + 0.04340277781 + t2 = 0.5 + 0.04340277781 Reorder the terms: 0.04340277781 + -0.4166666667t + t2 = 0.5 + 0.04340277781 Combine like terms: 0.5 + 0.04340277781 = 0.54340277781 0.04340277781 + -0.4166666667t + t2 = 0.54340277781 Factor a perfect square on the left side: (t + -0.2083333334)(t + -0.2083333334) = 0.54340277781 Calculate the square root of the right side: 0.737158584 Break this problem into two subproblems by setting (t + -0.2083333334) equal to 0.737158584 and -0.737158584.Subproblem 1
t + -0.2083333334 = 0.737158584 Simplifying t + -0.2083333334 = 0.737158584 Reorder the terms: -0.2083333334 + t = 0.737158584 Solving -0.2083333334 + t = 0.737158584 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.2083333334' to each side of the equation. -0.2083333334 + 0.2083333334 + t = 0.737158584 + 0.2083333334 Combine like terms: -0.2083333334 + 0.2083333334 = 0.0000000000 0.0000000000 + t = 0.737158584 + 0.2083333334 t = 0.737158584 + 0.2083333334 Combine like terms: 0.737158584 + 0.2083333334 = 0.9454919174 t = 0.9454919174 Simplifying t = 0.9454919174Subproblem 2
t + -0.2083333334 = -0.737158584 Simplifying t + -0.2083333334 = -0.737158584 Reorder the terms: -0.2083333334 + t = -0.737158584 Solving -0.2083333334 + t = -0.737158584 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.2083333334' to each side of the equation. -0.2083333334 + 0.2083333334 + t = -0.737158584 + 0.2083333334 Combine like terms: -0.2083333334 + 0.2083333334 = 0.0000000000 0.0000000000 + t = -0.737158584 + 0.2083333334 t = -0.737158584 + 0.2083333334 Combine like terms: -0.737158584 + 0.2083333334 = -0.5288252506 t = -0.5288252506 Simplifying t = -0.5288252506Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.9454919174, -0.5288252506}
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