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Simplifying 6t2 + t + -3 = 0 Reorder the terms: -3 + t + 6t2 = 0 Solving -3 + t + 6t2 = 0 Solving for variable 't'. Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -0.5 + 0.1666666667t + t2 = 0 Move the constant term to the right: Add '0.5' to each side of the equation. -0.5 + 0.1666666667t + 0.5 + t2 = 0 + 0.5 Reorder the terms: -0.5 + 0.5 + 0.1666666667t + t2 = 0 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + 0.1666666667t + t2 = 0 + 0.5 0.1666666667t + t2 = 0 + 0.5 Combine like terms: 0 + 0.5 = 0.5 0.1666666667t + t2 = 0.5 The t term is t. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. 0.1666666667t + 0.25 + t2 = 0.5 + 0.25 Reorder the terms: 0.25 + 0.1666666667t + t2 = 0.5 + 0.25 Combine like terms: 0.5 + 0.25 = 0.75 0.25 + 0.1666666667t + t2 = 0.75 Factor a perfect square on the left side: (t + 0.5)(t + 0.5) = 0.75 Calculate the square root of the right side: 0.866025404 Break this problem into two subproblems by setting (t + 0.5) equal to 0.866025404 and -0.866025404.Subproblem 1
t + 0.5 = 0.866025404 Simplifying t + 0.5 = 0.866025404 Reorder the terms: 0.5 + t = 0.866025404 Solving 0.5 + t = 0.866025404 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + t = 0.866025404 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + t = 0.866025404 + -0.5 t = 0.866025404 + -0.5 Combine like terms: 0.866025404 + -0.5 = 0.366025404 t = 0.366025404 Simplifying t = 0.366025404Subproblem 2
t + 0.5 = -0.866025404 Simplifying t + 0.5 = -0.866025404 Reorder the terms: 0.5 + t = -0.866025404 Solving 0.5 + t = -0.866025404 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + t = -0.866025404 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + t = -0.866025404 + -0.5 t = -0.866025404 + -0.5 Combine like terms: -0.866025404 + -0.5 = -1.366025404 t = -1.366025404 Simplifying t = -1.366025404Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.366025404, -1.366025404}
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