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Simplifying 0 = 3t2 + -6t + 2 Reorder the terms: 0 = 2 + -6t + 3t2 Solving 0 = 2 + -6t + 3t2 Solving for variable 't'. Combine like terms: 0 + -2 = -2 -2 + 6t + -3t2 = 2 + -6t + 3t2 + -2 + 6t + -3t2 Reorder the terms: -2 + 6t + -3t2 = 2 + -2 + -6t + 6t + 3t2 + -3t2 Combine like terms: 2 + -2 = 0 -2 + 6t + -3t2 = 0 + -6t + 6t + 3t2 + -3t2 -2 + 6t + -3t2 = -6t + 6t + 3t2 + -3t2 Combine like terms: -6t + 6t = 0 -2 + 6t + -3t2 = 0 + 3t2 + -3t2 -2 + 6t + -3t2 = 3t2 + -3t2 Combine like terms: 3t2 + -3t2 = 0 -2 + 6t + -3t2 = 0 Begin completing the square. Divide all terms by -3 the coefficient of the squared term: Divide each side by '-3'. 0.6666666667 + -2t + t2 = 0 Move the constant term to the right: Add '-0.6666666667' to each side of the equation. 0.6666666667 + -2t + -0.6666666667 + t2 = 0 + -0.6666666667 Reorder the terms: 0.6666666667 + -0.6666666667 + -2t + t2 = 0 + -0.6666666667 Combine like terms: 0.6666666667 + -0.6666666667 = 0.0000000000 0.0000000000 + -2t + t2 = 0 + -0.6666666667 -2t + t2 = 0 + -0.6666666667 Combine like terms: 0 + -0.6666666667 = -0.6666666667 -2t + t2 = -0.6666666667 The t term is -2t. Take half its coefficient (-1). Square it (1) and add it to both sides. Add '1' to each side of the equation. -2t + 1 + t2 = -0.6666666667 + 1 Reorder the terms: 1 + -2t + t2 = -0.6666666667 + 1 Combine like terms: -0.6666666667 + 1 = 0.3333333333 1 + -2t + t2 = 0.3333333333 Factor a perfect square on the left side: (t + -1)(t + -1) = 0.3333333333 Calculate the square root of the right side: 0.577350269 Break this problem into two subproblems by setting (t + -1) equal to 0.577350269 and -0.577350269.Subproblem 1
t + -1 = 0.577350269 Simplifying t + -1 = 0.577350269 Reorder the terms: -1 + t = 0.577350269 Solving -1 + t = 0.577350269 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + t = 0.577350269 + 1 Combine like terms: -1 + 1 = 0 0 + t = 0.577350269 + 1 t = 0.577350269 + 1 Combine like terms: 0.577350269 + 1 = 1.577350269 t = 1.577350269 Simplifying t = 1.577350269Subproblem 2
t + -1 = -0.577350269 Simplifying t + -1 = -0.577350269 Reorder the terms: -1 + t = -0.577350269 Solving -1 + t = -0.577350269 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + t = -0.577350269 + 1 Combine like terms: -1 + 1 = 0 0 + t = -0.577350269 + 1 t = -0.577350269 + 1 Combine like terms: -0.577350269 + 1 = 0.422649731 t = 0.422649731 Simplifying t = 0.422649731Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.577350269, 0.422649731}
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