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Simplifying 20 = 30t + -5t * t Multiply t * t 20 = 30t + -5t2 Solving 20 = 30t + -5t2 Solving for variable 't'. Reorder the terms: 20 + -30t + 5t2 = 30t + -30t + -5t2 + 5t2 Combine like terms: 30t + -30t = 0 20 + -30t + 5t2 = 0 + -5t2 + 5t2 20 + -30t + 5t2 = -5t2 + 5t2 Combine like terms: -5t2 + 5t2 = 0 20 + -30t + 5t2 = 0 Factor out the Greatest Common Factor (GCF), '5'. 5(4 + -6t + t2) = 0 Ignore the factor 5.Subproblem 1
Set the factor '(4 + -6t + t2)' equal to zero and attempt to solve: Simplifying 4 + -6t + t2 = 0 Solving 4 + -6t + t2 = 0 Begin completing the square. Move the constant term to the right: Add '-4' to each side of the equation. 4 + -6t + -4 + t2 = 0 + -4 Reorder the terms: 4 + -4 + -6t + t2 = 0 + -4 Combine like terms: 4 + -4 = 0 0 + -6t + t2 = 0 + -4 -6t + t2 = 0 + -4 Combine like terms: 0 + -4 = -4 -6t + t2 = -4 The t term is -6t. Take half its coefficient (-3). Square it (9) and add it to both sides. Add '9' to each side of the equation. -6t + 9 + t2 = -4 + 9 Reorder the terms: 9 + -6t + t2 = -4 + 9 Combine like terms: -4 + 9 = 5 9 + -6t + t2 = 5 Factor a perfect square on the left side: (t + -3)(t + -3) = 5 Calculate the square root of the right side: 2.236067978 Break this problem into two subproblems by setting (t + -3) equal to 2.236067978 and -2.236067978.Subproblem 1
t + -3 = 2.236067978 Simplifying t + -3 = 2.236067978 Reorder the terms: -3 + t = 2.236067978 Solving -3 + t = 2.236067978 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + t = 2.236067978 + 3 Combine like terms: -3 + 3 = 0 0 + t = 2.236067978 + 3 t = 2.236067978 + 3 Combine like terms: 2.236067978 + 3 = 5.236067978 t = 5.236067978 Simplifying t = 5.236067978Subproblem 2
t + -3 = -2.236067978 Simplifying t + -3 = -2.236067978 Reorder the terms: -3 + t = -2.236067978 Solving -3 + t = -2.236067978 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + t = -2.236067978 + 3 Combine like terms: -3 + 3 = 0 0 + t = -2.236067978 + 3 t = -2.236067978 + 3 Combine like terms: -2.236067978 + 3 = 0.763932022 t = 0.763932022 Simplifying t = 0.763932022Solution
The solution to the problem is based on the solutions from the subproblems. t = {5.236067978, 0.763932022}Solution
t = {5.236067978, 0.763932022}
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