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Simplifying 0 = -6t2 + 30t + -20 Reorder the terms: 0 = -20 + 30t + -6t2 Solving 0 = -20 + 30t + -6t2 Solving for variable 't'. Combine like terms: 0 + 20 = 20 20 + -30t + 6t2 = -20 + 30t + -6t2 + 20 + -30t + 6t2 Reorder the terms: 20 + -30t + 6t2 = -20 + 20 + 30t + -30t + -6t2 + 6t2 Combine like terms: -20 + 20 = 0 20 + -30t + 6t2 = 0 + 30t + -30t + -6t2 + 6t2 20 + -30t + 6t2 = 30t + -30t + -6t2 + 6t2 Combine like terms: 30t + -30t = 0 20 + -30t + 6t2 = 0 + -6t2 + 6t2 20 + -30t + 6t2 = -6t2 + 6t2 Combine like terms: -6t2 + 6t2 = 0 20 + -30t + 6t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(10 + -15t + 3t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(10 + -15t + 3t2)' equal to zero and attempt to solve: Simplifying 10 + -15t + 3t2 = 0 Solving 10 + -15t + 3t2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 3.333333333 + -5t + t2 = 0 Move the constant term to the right: Add '-3.333333333' to each side of the equation. 3.333333333 + -5t + -3.333333333 + t2 = 0 + -3.333333333 Reorder the terms: 3.333333333 + -3.333333333 + -5t + t2 = 0 + -3.333333333 Combine like terms: 3.333333333 + -3.333333333 = 0.000000000 0.000000000 + -5t + t2 = 0 + -3.333333333 -5t + t2 = 0 + -3.333333333 Combine like terms: 0 + -3.333333333 = -3.333333333 -5t + t2 = -3.333333333 The t term is -5t. Take half its coefficient (-2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. -5t + 6.25 + t2 = -3.333333333 + 6.25 Reorder the terms: 6.25 + -5t + t2 = -3.333333333 + 6.25 Combine like terms: -3.333333333 + 6.25 = 2.916666667 6.25 + -5t + t2 = 2.916666667 Factor a perfect square on the left side: (t + -2.5)(t + -2.5) = 2.916666667 Calculate the square root of the right side: 1.707825128 Break this problem into two subproblems by setting (t + -2.5) equal to 1.707825128 and -1.707825128.Subproblem 1
t + -2.5 = 1.707825128 Simplifying t + -2.5 = 1.707825128 Reorder the terms: -2.5 + t = 1.707825128 Solving -2.5 + t = 1.707825128 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + t = 1.707825128 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + t = 1.707825128 + 2.5 t = 1.707825128 + 2.5 Combine like terms: 1.707825128 + 2.5 = 4.207825128 t = 4.207825128 Simplifying t = 4.207825128Subproblem 2
t + -2.5 = -1.707825128 Simplifying t + -2.5 = -1.707825128 Reorder the terms: -2.5 + t = -1.707825128 Solving -2.5 + t = -1.707825128 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + t = -1.707825128 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + t = -1.707825128 + 2.5 t = -1.707825128 + 2.5 Combine like terms: -1.707825128 + 2.5 = 0.792174872 t = 0.792174872 Simplifying t = 0.792174872Solution
The solution to the problem is based on the solutions from the subproblems. t = {4.207825128, 0.792174872}Solution
t = {4.207825128, 0.792174872}
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