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Simplifying 40 = -16t2 + -48t + 160 Reorder the terms: 40 = 160 + -48t + -16t2 Solving 40 = 160 + -48t + -16t2 Solving for variable 't'. Combine like terms: 40 + -160 = -120 -120 + 48t + 16t2 = 160 + -48t + -16t2 + -160 + 48t + 16t2 Reorder the terms: -120 + 48t + 16t2 = 160 + -160 + -48t + 48t + -16t2 + 16t2 Combine like terms: 160 + -160 = 0 -120 + 48t + 16t2 = 0 + -48t + 48t + -16t2 + 16t2 -120 + 48t + 16t2 = -48t + 48t + -16t2 + 16t2 Combine like terms: -48t + 48t = 0 -120 + 48t + 16t2 = 0 + -16t2 + 16t2 -120 + 48t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -120 + 48t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '8'. 8(-15 + 6t + 2t2) = 0 Ignore the factor 8.Subproblem 1
Set the factor '(-15 + 6t + 2t2)' equal to zero and attempt to solve: Simplifying -15 + 6t + 2t2 = 0 Solving -15 + 6t + 2t2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -7.5 + 3t + t2 = 0 Move the constant term to the right: Add '7.5' to each side of the equation. -7.5 + 3t + 7.5 + t2 = 0 + 7.5 Reorder the terms: -7.5 + 7.5 + 3t + t2 = 0 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + 3t + t2 = 0 + 7.5 3t + t2 = 0 + 7.5 Combine like terms: 0 + 7.5 = 7.5 3t + t2 = 7.5 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 = 7.5 + 2.25 Reorder the terms: 2.25 + 3t + t2 = 7.5 + 2.25 Combine like terms: 7.5 + 2.25 = 9.75 2.25 + 3t + t2 = 9.75 Factor a perfect square on the left side: (t + 1.5)(t + 1.5) = 9.75 Calculate the square root of the right side: 3.122498999 Break this problem into two subproblems by setting (t + 1.5) equal to 3.122498999 and -3.122498999.Subproblem 1
t + 1.5 = 3.122498999 Simplifying t + 1.5 = 3.122498999 Reorder the terms: 1.5 + t = 3.122498999 Solving 1.5 + t = 3.122498999 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 = 3.122498999 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + t = 3.122498999 + -1.5 t = 3.122498999 + -1.5 Combine like terms: 3.122498999 + -1.5 = 1.622498999 t = 1.622498999 Simplifying t = 1.622498999Subproblem 2
t + 1.5 = -3.122498999 Simplifying t + 1.5 = -3.122498999 Reorder the terms: 1.5 + t = -3.122498999 Solving 1.5 + t = -3.122498999 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 = -3.122498999 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + t = -3.122498999 + -1.5 t = -3.122498999 + -1.5 Combine like terms: -3.122498999 + -1.5 = -4.622498999 t = -4.622498999 Simplifying t = -4.622498999Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.622498999, -4.622498999}Solution
t = {1.622498999, -4.622498999}
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