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Simplifying 6t2 = 18t + -8 Reorder the terms: 6t2 = -8 + 18t Solving 6t2 = -8 + 18t Solving for variable 't'. Reorder the terms: 8 + -18t + 6t2 = -8 + 18t + 8 + -18t Reorder the terms: 8 + -18t + 6t2 = -8 + 8 + 18t + -18t Combine like terms: -8 + 8 = 0 8 + -18t + 6t2 = 0 + 18t + -18t 8 + -18t + 6t2 = 18t + -18t Combine like terms: 18t + -18t = 0 8 + -18t + 6t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(4 + -9t + 3t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(4 + -9t + 3t2)' equal to zero and attempt to solve: Simplifying 4 + -9t + 3t2 = 0 Solving 4 + -9t + 3t2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 1.333333333 + -3t + t2 = 0 Move the constant term to the right: Add '-1.333333333' to each side of the equation. 1.333333333 + -3t + -1.333333333 + t2 = 0 + -1.333333333 Reorder the terms: 1.333333333 + -1.333333333 + -3t + t2 = 0 + -1.333333333 Combine like terms: 1.333333333 + -1.333333333 = 0.000000000 0.000000000 + -3t + t2 = 0 + -1.333333333 -3t + t2 = 0 + -1.333333333 Combine like terms: 0 + -1.333333333 = -1.333333333 -3t + t2 = -1.333333333 The t term is -3t. Take half its coefficient (-1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. -3t + 2.25 + t2 = -1.333333333 + 2.25 Reorder the terms: 2.25 + -3t + t2 = -1.333333333 + 2.25 Combine like terms: -1.333333333 + 2.25 = 0.916666667 2.25 + -3t + t2 = 0.916666667 Factor a perfect square on the left side: (t + -1.5)(t + -1.5) = 0.916666667 Calculate the square root of the right side: 0.957427108 Break this problem into two subproblems by setting (t + -1.5) equal to 0.957427108 and -0.957427108.Subproblem 1
t + -1.5 = 0.957427108 Simplifying t + -1.5 = 0.957427108 Reorder the terms: -1.5 + t = 0.957427108 Solving -1.5 + t = 0.957427108 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + t = 0.957427108 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + t = 0.957427108 + 1.5 t = 0.957427108 + 1.5 Combine like terms: 0.957427108 + 1.5 = 2.457427108 t = 2.457427108 Simplifying t = 2.457427108Subproblem 2
t + -1.5 = -0.957427108 Simplifying t + -1.5 = -0.957427108 Reorder the terms: -1.5 + t = -0.957427108 Solving -1.5 + t = -0.957427108 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + t = -0.957427108 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + t = -0.957427108 + 1.5 t = -0.957427108 + 1.5 Combine like terms: -0.957427108 + 1.5 = 0.542572892 t = 0.542572892 Simplifying t = 0.542572892Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.457427108, 0.542572892}Solution
t = {2.457427108, 0.542572892}
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