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Simplifying 6t2 + 19t = 6 Reorder the terms: 19t + 6t2 = 6 Solving 19t + 6t2 = 6 Solving for variable 't'. Reorder the terms: -6 + 19t + 6t2 = 6 + -6 Combine like terms: 6 + -6 = 0 -6 + 19t + 6t2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -1 + 3.166666667t + t2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + 3.166666667t + 1 + t2 = 0 + 1 Reorder the terms: -1 + 1 + 3.166666667t + t2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + 3.166666667t + t2 = 0 + 1 3.166666667t + t2 = 0 + 1 Combine like terms: 0 + 1 = 1 3.166666667t + t2 = 1 The t term is 3.166666667t. Take half its coefficient (1.583333334). Square it (2.506944447) and add it to both sides. Add '2.506944447' to each side of the equation. 3.166666667t + 2.506944447 + t2 = 1 + 2.506944447 Reorder the terms: 2.506944447 + 3.166666667t + t2 = 1 + 2.506944447 Combine like terms: 1 + 2.506944447 = 3.506944447 2.506944447 + 3.166666667t + t2 = 3.506944447 Factor a perfect square on the left side: (t + 1.583333334)(t + 1.583333334) = 3.506944447 Calculate the square root of the right side: 1.872683755 Break this problem into two subproblems by setting (t + 1.583333334) equal to 1.872683755 and -1.872683755.Subproblem 1
t + 1.583333334 = 1.872683755 Simplifying t + 1.583333334 = 1.872683755 Reorder the terms: 1.583333334 + t = 1.872683755 Solving 1.583333334 + t = 1.872683755 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1.583333334' to each side of the equation. 1.583333334 + -1.583333334 + t = 1.872683755 + -1.583333334 Combine like terms: 1.583333334 + -1.583333334 = 0.000000000 0.000000000 + t = 1.872683755 + -1.583333334 t = 1.872683755 + -1.583333334 Combine like terms: 1.872683755 + -1.583333334 = 0.289350421 t = 0.289350421 Simplifying t = 0.289350421Subproblem 2
t + 1.583333334 = -1.872683755 Simplifying t + 1.583333334 = -1.872683755 Reorder the terms: 1.583333334 + t = -1.872683755 Solving 1.583333334 + t = -1.872683755 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1.583333334' to each side of the equation. 1.583333334 + -1.583333334 + t = -1.872683755 + -1.583333334 Combine like terms: 1.583333334 + -1.583333334 = 0.000000000 0.000000000 + t = -1.872683755 + -1.583333334 t = -1.872683755 + -1.583333334 Combine like terms: -1.872683755 + -1.583333334 = -3.456017089 t = -3.456017089 Simplifying t = -3.456017089Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.289350421, -3.456017089}
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