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Simplifying 0 = -16t2 + 18t + 6 Reorder the terms: 0 = 6 + 18t + -16t2 Solving 0 = 6 + 18t + -16t2 Solving for variable 't'. Combine like terms: 0 + -6 = -6 -6 + -18t + 16t2 = 6 + 18t + -16t2 + -6 + -18t + 16t2 Reorder the terms: -6 + -18t + 16t2 = 6 + -6 + 18t + -18t + -16t2 + 16t2 Combine like terms: 6 + -6 = 0 -6 + -18t + 16t2 = 0 + 18t + -18t + -16t2 + 16t2 -6 + -18t + 16t2 = 18t + -18t + -16t2 + 16t2 Combine like terms: 18t + -18t = 0 -6 + -18t + 16t2 = 0 + -16t2 + 16t2 -6 + -18t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -6 + -18t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-3 + -9t + 8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-3 + -9t + 8t2)' equal to zero and attempt to solve: Simplifying -3 + -9t + 8t2 = 0 Solving -3 + -9t + 8t2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -0.375 + -1.125t + t2 = 0 Move the constant term to the right: Add '0.375' to each side of the equation. -0.375 + -1.125t + 0.375 + t2 = 0 + 0.375 Reorder the terms: -0.375 + 0.375 + -1.125t + t2 = 0 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + -1.125t + t2 = 0 + 0.375 -1.125t + t2 = 0 + 0.375 Combine like terms: 0 + 0.375 = 0.375 -1.125t + t2 = 0.375 The t term is -1.125t. Take half its coefficient (-0.5625). Square it (0.31640625) and add it to both sides. Add '0.31640625' to each side of the equation. -1.125t + 0.31640625 + t2 = 0.375 + 0.31640625 Reorder the terms: 0.31640625 + -1.125t + t2 = 0.375 + 0.31640625 Combine like terms: 0.375 + 0.31640625 = 0.69140625 0.31640625 + -1.125t + t2 = 0.69140625 Factor a perfect square on the left side: (t + -0.5625)(t + -0.5625) = 0.69140625 Calculate the square root of the right side: 0.831508418 Break this problem into two subproblems by setting (t + -0.5625) equal to 0.831508418 and -0.831508418.Subproblem 1
t + -0.5625 = 0.831508418 Simplifying t + -0.5625 = 0.831508418 Reorder the terms: -0.5625 + t = 0.831508418 Solving -0.5625 + t = 0.831508418 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.5625' to each side of the equation. -0.5625 + 0.5625 + t = 0.831508418 + 0.5625 Combine like terms: -0.5625 + 0.5625 = 0.0000 0.0000 + t = 0.831508418 + 0.5625 t = 0.831508418 + 0.5625 Combine like terms: 0.831508418 + 0.5625 = 1.394008418 t = 1.394008418 Simplifying t = 1.394008418Subproblem 2
t + -0.5625 = -0.831508418 Simplifying t + -0.5625 = -0.831508418 Reorder the terms: -0.5625 + t = -0.831508418 Solving -0.5625 + t = -0.831508418 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.5625' to each side of the equation. -0.5625 + 0.5625 + t = -0.831508418 + 0.5625 Combine like terms: -0.5625 + 0.5625 = 0.0000 0.0000 + t = -0.831508418 + 0.5625 t = -0.831508418 + 0.5625 Combine like terms: -0.831508418 + 0.5625 = -0.269008418 t = -0.269008418 Simplifying t = -0.269008418Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.394008418, -0.269008418}Solution
t = {1.394008418, -0.269008418}
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