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Simplifying 0 = 6t2 + -22t + 3 Reorder the terms: 0 = 3 + -22t + 6t2 Solving 0 = 3 + -22t + 6t2 Solving for variable 't'. Combine like terms: 0 + -3 = -3 -3 + 22t + -6t2 = 3 + -22t + 6t2 + -3 + 22t + -6t2 Reorder the terms: -3 + 22t + -6t2 = 3 + -3 + -22t + 22t + 6t2 + -6t2 Combine like terms: 3 + -3 = 0 -3 + 22t + -6t2 = 0 + -22t + 22t + 6t2 + -6t2 -3 + 22t + -6t2 = -22t + 22t + 6t2 + -6t2 Combine like terms: -22t + 22t = 0 -3 + 22t + -6t2 = 0 + 6t2 + -6t2 -3 + 22t + -6t2 = 6t2 + -6t2 Combine like terms: 6t2 + -6t2 = 0 -3 + 22t + -6t2 = 0 Begin completing the square. Divide all terms by -6 the coefficient of the squared term: Divide each side by '-6'. 0.5 + -3.666666667t + t2 = 0 Move the constant term to the right: Add '-0.5' to each side of the equation. 0.5 + -3.666666667t + -0.5 + t2 = 0 + -0.5 Reorder the terms: 0.5 + -0.5 + -3.666666667t + t2 = 0 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + -3.666666667t + t2 = 0 + -0.5 -3.666666667t + t2 = 0 + -0.5 Combine like terms: 0 + -0.5 = -0.5 -3.666666667t + t2 = -0.5 The t term is -3.666666667t. Take half its coefficient (-1.833333334). Square it (3.361111114) and add it to both sides. Add '3.361111114' to each side of the equation. -3.666666667t + 3.361111114 + t2 = -0.5 + 3.361111114 Reorder the terms: 3.361111114 + -3.666666667t + t2 = -0.5 + 3.361111114 Combine like terms: -0.5 + 3.361111114 = 2.861111114 3.361111114 + -3.666666667t + t2 = 2.861111114 Factor a perfect square on the left side: (t + -1.833333334)(t + -1.833333334) = 2.861111114 Calculate the square root of the right side: 1.691481928 Break this problem into two subproblems by setting (t + -1.833333334) equal to 1.691481928 and -1.691481928.Subproblem 1
t + -1.833333334 = 1.691481928 Simplifying t + -1.833333334 = 1.691481928 Reorder the terms: -1.833333334 + t = 1.691481928 Solving -1.833333334 + t = 1.691481928 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.833333334' to each side of the equation. -1.833333334 + 1.833333334 + t = 1.691481928 + 1.833333334 Combine like terms: -1.833333334 + 1.833333334 = 0.000000000 0.000000000 + t = 1.691481928 + 1.833333334 t = 1.691481928 + 1.833333334 Combine like terms: 1.691481928 + 1.833333334 = 3.524815262 t = 3.524815262 Simplifying t = 3.524815262Subproblem 2
t + -1.833333334 = -1.691481928 Simplifying t + -1.833333334 = -1.691481928 Reorder the terms: -1.833333334 + t = -1.691481928 Solving -1.833333334 + t = -1.691481928 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.833333334' to each side of the equation. -1.833333334 + 1.833333334 + t = -1.691481928 + 1.833333334 Combine like terms: -1.833333334 + 1.833333334 = 0.000000000 0.000000000 + t = -1.691481928 + 1.833333334 t = -1.691481928 + 1.833333334 Combine like terms: -1.691481928 + 1.833333334 = 0.141851406 t = 0.141851406 Simplifying t = 0.141851406Solution
The solution to the problem is based on the solutions from the subproblems. t = {3.524815262, 0.141851406}
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