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Simplifying 1000 = 256t + -16t2 Solving 1000 = 256t + -16t2 Solving for variable 't'. Reorder the terms: 1000 + -256t + 16t2 = 256t + -256t + -16t2 + 16t2 Combine like terms: 256t + -256t = 0 1000 + -256t + 16t2 = 0 + -16t2 + 16t2 1000 + -256t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 1000 + -256t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '8'. 8(125 + -32t + 2t2) = 0 Ignore the factor 8.Subproblem 1
Set the factor '(125 + -32t + 2t2)' equal to zero and attempt to solve: Simplifying 125 + -32t + 2t2 = 0 Solving 125 + -32t + 2t2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 62.5 + -16t + t2 = 0 Move the constant term to the right: Add '-62.5' to each side of the equation. 62.5 + -16t + -62.5 + t2 = 0 + -62.5 Reorder the terms: 62.5 + -62.5 + -16t + t2 = 0 + -62.5 Combine like terms: 62.5 + -62.5 = 0.0 0.0 + -16t + t2 = 0 + -62.5 -16t + t2 = 0 + -62.5 Combine like terms: 0 + -62.5 = -62.5 -16t + t2 = -62.5 The t term is -16t. Take half its coefficient (-8). Square it (64) and add it to both sides. Add '64' to each side of the equation. -16t + 64 + t2 = -62.5 + 64 Reorder the terms: 64 + -16t + t2 = -62.5 + 64 Combine like terms: -62.5 + 64 = 1.5 64 + -16t + t2 = 1.5 Factor a perfect square on the left side: (t + -8)(t + -8) = 1.5 Calculate the square root of the right side: 1.224744871 Break this problem into two subproblems by setting (t + -8) equal to 1.224744871 and -1.224744871.Subproblem 1
t + -8 = 1.224744871 Simplifying t + -8 = 1.224744871 Reorder the terms: -8 + t = 1.224744871 Solving -8 + t = 1.224744871 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '8' to each side of the equation. -8 + 8 + t = 1.224744871 + 8 Combine like terms: -8 + 8 = 0 0 + t = 1.224744871 + 8 t = 1.224744871 + 8 Combine like terms: 1.224744871 + 8 = 9.224744871 t = 9.224744871 Simplifying t = 9.224744871Subproblem 2
t + -8 = -1.224744871 Simplifying t + -8 = -1.224744871 Reorder the terms: -8 + t = -1.224744871 Solving -8 + t = -1.224744871 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '8' to each side of the equation. -8 + 8 + t = -1.224744871 + 8 Combine like terms: -8 + 8 = 0 0 + t = -1.224744871 + 8 t = -1.224744871 + 8 Combine like terms: -1.224744871 + 8 = 6.775255129 t = 6.775255129 Simplifying t = 6.775255129Solution
The solution to the problem is based on the solutions from the subproblems. t = {9.224744871, 6.775255129}Solution
t = {9.224744871, 6.775255129}
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