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Simplifying 3t2 + 8t + -6 = 0 Reorder the terms: -6 + 8t + 3t2 = 0 Solving -6 + 8t + 3t2 = 0 Solving for variable 't'. Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -2 + 2.666666667t + t2 = 0 Move the constant term to the right: Add '2' to each side of the equation. -2 + 2.666666667t + 2 + t2 = 0 + 2 Reorder the terms: -2 + 2 + 2.666666667t + t2 = 0 + 2 Combine like terms: -2 + 2 = 0 0 + 2.666666667t + t2 = 0 + 2 2.666666667t + t2 = 0 + 2 Combine like terms: 0 + 2 = 2 2.666666667t + t2 = 2 The t term is 2.666666667t. Take half its coefficient (1.333333334). Square it (1.777777780) and add it to both sides. Add '1.777777780' to each side of the equation. 2.666666667t + 1.777777780 + t2 = 2 + 1.777777780 Reorder the terms: 1.777777780 + 2.666666667t + t2 = 2 + 1.777777780 Combine like terms: 2 + 1.777777780 = 3.77777778 1.777777780 + 2.666666667t + t2 = 3.77777778 Factor a perfect square on the left side: (t + 1.333333334)(t + 1.333333334) = 3.77777778 Calculate the square root of the right side: 1.943650632 Break this problem into two subproblems by setting (t + 1.333333334) equal to 1.943650632 and -1.943650632.Subproblem 1
t + 1.333333334 = 1.943650632 Simplifying t + 1.333333334 = 1.943650632 Reorder the terms: 1.333333334 + t = 1.943650632 Solving 1.333333334 + t = 1.943650632 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1.333333334' to each side of the equation. 1.333333334 + -1.333333334 + t = 1.943650632 + -1.333333334 Combine like terms: 1.333333334 + -1.333333334 = 0.000000000 0.000000000 + t = 1.943650632 + -1.333333334 t = 1.943650632 + -1.333333334 Combine like terms: 1.943650632 + -1.333333334 = 0.610317298 t = 0.610317298 Simplifying t = 0.610317298Subproblem 2
t + 1.333333334 = -1.943650632 Simplifying t + 1.333333334 = -1.943650632 Reorder the terms: 1.333333334 + t = -1.943650632 Solving 1.333333334 + t = -1.943650632 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1.333333334' to each side of the equation. 1.333333334 + -1.333333334 + t = -1.943650632 + -1.333333334 Combine like terms: 1.333333334 + -1.333333334 = 0.000000000 0.000000000 + t = -1.943650632 + -1.333333334 t = -1.943650632 + -1.333333334 Combine like terms: -1.943650632 + -1.333333334 = -3.276983966 t = -3.276983966 Simplifying t = -3.276983966Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.610317298, -3.276983966}
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