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Simplifying 8 = 22t + -6t2 Solving 8 = 22t + -6t2 Solving for variable 't'. Reorder the terms: 8 + -22t + 6t2 = 22t + -22t + -6t2 + 6t2 Combine like terms: 22t + -22t = 0 8 + -22t + 6t2 = 0 + -6t2 + 6t2 8 + -22t + 6t2 = -6t2 + 6t2 Combine like terms: -6t2 + 6t2 = 0 8 + -22t + 6t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(4 + -11t + 3t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(4 + -11t + 3t2)' equal to zero and attempt to solve: Simplifying 4 + -11t + 3t2 = 0 Solving 4 + -11t + 3t2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 1.333333333 + -3.666666667t + t2 = 0 Move the constant term to the right: Add '-1.333333333' to each side of the equation. 1.333333333 + -3.666666667t + -1.333333333 + t2 = 0 + -1.333333333 Reorder the terms: 1.333333333 + -1.333333333 + -3.666666667t + t2 = 0 + -1.333333333 Combine like terms: 1.333333333 + -1.333333333 = 0.000000000 0.000000000 + -3.666666667t + t2 = 0 + -1.333333333 -3.666666667t + t2 = 0 + -1.333333333 Combine like terms: 0 + -1.333333333 = -1.333333333 -3.666666667t + t2 = -1.333333333 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 = -1.333333333 + 3.361111114 Reorder the terms: 3.361111114 + -3.666666667t + t2 = -1.333333333 + 3.361111114 Combine like terms: -1.333333333 + 3.361111114 = 2.027777781 3.361111114 + -3.666666667t + t2 = 2.027777781 Factor a perfect square on the left side: (t + -1.833333334)(t + -1.833333334) = 2.027777781 Calculate the square root of the right side: 1.424000625 Break this problem into two subproblems by setting (t + -1.833333334) equal to 1.424000625 and -1.424000625.Subproblem 1
t + -1.833333334 = 1.424000625 Simplifying t + -1.833333334 = 1.424000625 Reorder the terms: -1.833333334 + t = 1.424000625 Solving -1.833333334 + t = 1.424000625 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 = 1.424000625 + 1.833333334 Combine like terms: -1.833333334 + 1.833333334 = 0.000000000 0.000000000 + t = 1.424000625 + 1.833333334 t = 1.424000625 + 1.833333334 Combine like terms: 1.424000625 + 1.833333334 = 3.257333959 t = 3.257333959 Simplifying t = 3.257333959Subproblem 2
t + -1.833333334 = -1.424000625 Simplifying t + -1.833333334 = -1.424000625 Reorder the terms: -1.833333334 + t = -1.424000625 Solving -1.833333334 + t = -1.424000625 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 = -1.424000625 + 1.833333334 Combine like terms: -1.833333334 + 1.833333334 = 0.000000000 0.000000000 + t = -1.424000625 + 1.833333334 t = -1.424000625 + 1.833333334 Combine like terms: -1.424000625 + 1.833333334 = 0.409332709 t = 0.409332709 Simplifying t = 0.409332709Solution
The solution to the problem is based on the solutions from the subproblems. t = {3.257333959, 0.409332709}Solution
t = {3.257333959, 0.409332709}
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