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Simplifying -6t2 + 22t + 21 = 0 Reorder the terms: 21 + 22t + -6t2 = 0 Solving 21 + 22t + -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'. -3.5 + -3.666666667t + t2 = 0 Move the constant term to the right: Add '3.5' to each side of the equation. -3.5 + -3.666666667t + 3.5 + t2 = 0 + 3.5 Reorder the terms: -3.5 + 3.5 + -3.666666667t + t2 = 0 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + -3.666666667t + t2 = 0 + 3.5 -3.666666667t + t2 = 0 + 3.5 Combine like terms: 0 + 3.5 = 3.5 -3.666666667t + t2 = 3.5 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 = 3.5 + 3.361111114 Reorder the terms: 3.361111114 + -3.666666667t + t2 = 3.5 + 3.361111114 Combine like terms: 3.5 + 3.361111114 = 6.861111114 3.361111114 + -3.666666667t + t2 = 6.861111114 Factor a perfect square on the left side: (t + -1.833333334)(t + -1.833333334) = 6.861111114 Calculate the square root of the right side: 2.619372275 Break this problem into two subproblems by setting (t + -1.833333334) equal to 2.619372275 and -2.619372275.Subproblem 1
t + -1.833333334 = 2.619372275 Simplifying t + -1.833333334 = 2.619372275 Reorder the terms: -1.833333334 + t = 2.619372275 Solving -1.833333334 + t = 2.619372275 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 = 2.619372275 + 1.833333334 Combine like terms: -1.833333334 + 1.833333334 = 0.000000000 0.000000000 + t = 2.619372275 + 1.833333334 t = 2.619372275 + 1.833333334 Combine like terms: 2.619372275 + 1.833333334 = 4.452705609 t = 4.452705609 Simplifying t = 4.452705609Subproblem 2
t + -1.833333334 = -2.619372275 Simplifying t + -1.833333334 = -2.619372275 Reorder the terms: -1.833333334 + t = -2.619372275 Solving -1.833333334 + t = -2.619372275 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 = -2.619372275 + 1.833333334 Combine like terms: -1.833333334 + 1.833333334 = 0.000000000 0.000000000 + t = -2.619372275 + 1.833333334 t = -2.619372275 + 1.833333334 Combine like terms: -2.619372275 + 1.833333334 = -0.786038941 t = -0.786038941 Simplifying t = -0.786038941Solution
The solution to the problem is based on the solutions from the subproblems. t = {4.452705609, -0.786038941}
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