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Simplifying 30 = -16t2 + -48t + 160 Reorder the terms: 30 = 160 + -48t + -16t2 Solving 30 = 160 + -48t + -16t2 Solving for variable 't'. Combine like terms: 30 + -160 = -130 -130 + 48t + 16t2 = 160 + -48t + -16t2 + -160 + 48t + 16t2 Reorder the terms: -130 + 48t + 16t2 = 160 + -160 + -48t + 48t + -16t2 + 16t2 Combine like terms: 160 + -160 = 0 -130 + 48t + 16t2 = 0 + -48t + 48t + -16t2 + 16t2 -130 + 48t + 16t2 = -48t + 48t + -16t2 + 16t2 Combine like terms: -48t + 48t = 0 -130 + 48t + 16t2 = 0 + -16t2 + 16t2 -130 + 48t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -130 + 48t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-65 + 24t + 8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-65 + 24t + 8t2)' equal to zero and attempt to solve: Simplifying -65 + 24t + 8t2 = 0 Solving -65 + 24t + 8t2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -8.125 + 3t + t2 = 0 Move the constant term to the right: Add '8.125' to each side of the equation. -8.125 + 3t + 8.125 + t2 = 0 + 8.125 Reorder the terms: -8.125 + 8.125 + 3t + t2 = 0 + 8.125 Combine like terms: -8.125 + 8.125 = 0.000 0.000 + 3t + t2 = 0 + 8.125 3t + t2 = 0 + 8.125 Combine like terms: 0 + 8.125 = 8.125 3t + t2 = 8.125 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 = 8.125 + 2.25 Reorder the terms: 2.25 + 3t + t2 = 8.125 + 2.25 Combine like terms: 8.125 + 2.25 = 10.375 2.25 + 3t + t2 = 10.375 Factor a perfect square on the left side: (t + 1.5)(t + 1.5) = 10.375 Calculate the square root of the right side: 3.221024682 Break this problem into two subproblems by setting (t + 1.5) equal to 3.221024682 and -3.221024682.Subproblem 1
t + 1.5 = 3.221024682 Simplifying t + 1.5 = 3.221024682 Reorder the terms: 1.5 + t = 3.221024682 Solving 1.5 + t = 3.221024682 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 = 3.221024682 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + t = 3.221024682 + -1.5 t = 3.221024682 + -1.5 Combine like terms: 3.221024682 + -1.5 = 1.721024682 t = 1.721024682 Simplifying t = 1.721024682Subproblem 2
t + 1.5 = -3.221024682 Simplifying t + 1.5 = -3.221024682 Reorder the terms: 1.5 + t = -3.221024682 Solving 1.5 + t = -3.221024682 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 = -3.221024682 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + t = -3.221024682 + -1.5 t = -3.221024682 + -1.5 Combine like terms: -3.221024682 + -1.5 = -4.721024682 t = -4.721024682 Simplifying t = -4.721024682Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.721024682, -4.721024682}Solution
t = {1.721024682, -4.721024682}
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