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Simplifying 6t2 + 13t + -6 = 0 Reorder the terms: -6 + 13t + 6t2 = 0 Solving -6 + 13t + 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'. -1 + 2.166666667t + t2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + 2.166666667t + 1 + t2 = 0 + 1 Reorder the terms: -1 + 1 + 2.166666667t + t2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + 2.166666667t + t2 = 0 + 1 2.166666667t + t2 = 0 + 1 Combine like terms: 0 + 1 = 1 2.166666667t + t2 = 1 The t term is 2.166666667t. Take half its coefficient (1.083333334). Square it (1.173611113) and add it to both sides. Add '1.173611113' to each side of the equation. 2.166666667t + 1.173611113 + t2 = 1 + 1.173611113 Reorder the terms: 1.173611113 + 2.166666667t + t2 = 1 + 1.173611113 Combine like terms: 1 + 1.173611113 = 2.173611113 1.173611113 + 2.166666667t + t2 = 2.173611113 Factor a perfect square on the left side: (t + 1.083333334)(t + 1.083333334) = 2.173611113 Calculate the square root of the right side: 1.474317168 Break this problem into two subproblems by setting (t + 1.083333334) equal to 1.474317168 and -1.474317168.Subproblem 1
t + 1.083333334 = 1.474317168 Simplifying t + 1.083333334 = 1.474317168 Reorder the terms: 1.083333334 + t = 1.474317168 Solving 1.083333334 + t = 1.474317168 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1.083333334' to each side of the equation. 1.083333334 + -1.083333334 + t = 1.474317168 + -1.083333334 Combine like terms: 1.083333334 + -1.083333334 = 0.000000000 0.000000000 + t = 1.474317168 + -1.083333334 t = 1.474317168 + -1.083333334 Combine like terms: 1.474317168 + -1.083333334 = 0.390983834 t = 0.390983834 Simplifying t = 0.390983834Subproblem 2
t + 1.083333334 = -1.474317168 Simplifying t + 1.083333334 = -1.474317168 Reorder the terms: 1.083333334 + t = -1.474317168 Solving 1.083333334 + t = -1.474317168 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1.083333334' to each side of the equation. 1.083333334 + -1.083333334 + t = -1.474317168 + -1.083333334 Combine like terms: 1.083333334 + -1.083333334 = 0.000000000 0.000000000 + t = -1.474317168 + -1.083333334 t = -1.474317168 + -1.083333334 Combine like terms: -1.474317168 + -1.083333334 = -2.557650502 t = -2.557650502 Simplifying t = -2.557650502Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.390983834, -2.557650502}
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