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Simplifying 500(t) = 16t2 + 500 Reorder the terms: 500t = 500 + 16t2 Solving 500t = 500 + 16t2 Solving for variable 't'. Reorder the terms: -500 + 500t + -16t2 = 500 + 16t2 + -500 + -16t2 Reorder the terms: -500 + 500t + -16t2 = 500 + -500 + 16t2 + -16t2 Combine like terms: 500 + -500 = 0 -500 + 500t + -16t2 = 0 + 16t2 + -16t2 -500 + 500t + -16t2 = 16t2 + -16t2 Combine like terms: 16t2 + -16t2 = 0 -500 + 500t + -16t2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(-125 + 125t + -4t2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(-125 + 125t + -4t2)' equal to zero and attempt to solve: Simplifying -125 + 125t + -4t2 = 0 Solving -125 + 125t + -4t2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. 31.25 + -31.25t + t2 = 0 Move the constant term to the right: Add '-31.25' to each side of the equation. 31.25 + -31.25t + -31.25 + t2 = 0 + -31.25 Reorder the terms: 31.25 + -31.25 + -31.25t + t2 = 0 + -31.25 Combine like terms: 31.25 + -31.25 = 0.00 0.00 + -31.25t + t2 = 0 + -31.25 -31.25t + t2 = 0 + -31.25 Combine like terms: 0 + -31.25 = -31.25 -31.25t + t2 = -31.25 The t term is -31.25t. Take half its coefficient (-15.625). Square it (244.140625) and add it to both sides. Add '244.140625' to each side of the equation. -31.25t + 244.140625 + t2 = -31.25 + 244.140625 Reorder the terms: 244.140625 + -31.25t + t2 = -31.25 + 244.140625 Combine like terms: -31.25 + 244.140625 = 212.890625 244.140625 + -31.25t + t2 = 212.890625 Factor a perfect square on the left side: (t + -15.625)(t + -15.625) = 212.890625 Calculate the square root of the right side: 14.590771912 Break this problem into two subproblems by setting (t + -15.625) equal to 14.590771912 and -14.590771912.Subproblem 1
t + -15.625 = 14.590771912 Simplifying t + -15.625 = 14.590771912 Reorder the terms: -15.625 + t = 14.590771912 Solving -15.625 + t = 14.590771912 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '15.625' to each side of the equation. -15.625 + 15.625 + t = 14.590771912 + 15.625 Combine like terms: -15.625 + 15.625 = 0.000 0.000 + t = 14.590771912 + 15.625 t = 14.590771912 + 15.625 Combine like terms: 14.590771912 + 15.625 = 30.215771912 t = 30.215771912 Simplifying t = 30.215771912Subproblem 2
t + -15.625 = -14.590771912 Simplifying t + -15.625 = -14.590771912 Reorder the terms: -15.625 + t = -14.590771912 Solving -15.625 + t = -14.590771912 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '15.625' to each side of the equation. -15.625 + 15.625 + t = -14.590771912 + 15.625 Combine like terms: -15.625 + 15.625 = 0.000 0.000 + t = -14.590771912 + 15.625 t = -14.590771912 + 15.625 Combine like terms: -14.590771912 + 15.625 = 1.034228088 t = 1.034228088 Simplifying t = 1.034228088Solution
The solution to the problem is based on the solutions from the subproblems. t = {30.215771912, 1.034228088}Solution
t = {30.215771912, 1.034228088}
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