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Simplifying 30 = -16t2 + 50t + 6 Reorder the terms: 30 = 6 + 50t + -16t2 Solving 30 = 6 + 50t + -16t2 Solving for variable 't'. Combine like terms: 30 + -6 = 24 24 + -50t + 16t2 = 6 + 50t + -16t2 + -6 + -50t + 16t2 Reorder the terms: 24 + -50t + 16t2 = 6 + -6 + 50t + -50t + -16t2 + 16t2 Combine like terms: 6 + -6 = 0 24 + -50t + 16t2 = 0 + 50t + -50t + -16t2 + 16t2 24 + -50t + 16t2 = 50t + -50t + -16t2 + 16t2 Combine like terms: 50t + -50t = 0 24 + -50t + 16t2 = 0 + -16t2 + 16t2 24 + -50t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 24 + -50t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(12 + -25t + 8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(12 + -25t + 8t2)' equal to zero and attempt to solve: Simplifying 12 + -25t + 8t2 = 0 Solving 12 + -25t + 8t2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. 1.5 + -3.125t + t2 = 0 Move the constant term to the right: Add '-1.5' to each side of the equation. 1.5 + -3.125t + -1.5 + t2 = 0 + -1.5 Reorder the terms: 1.5 + -1.5 + -3.125t + t2 = 0 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + -3.125t + t2 = 0 + -1.5 -3.125t + t2 = 0 + -1.5 Combine like terms: 0 + -1.5 = -1.5 -3.125t + t2 = -1.5 The t term is -3.125t. Take half its coefficient (-1.5625). Square it (2.44140625) and add it to both sides. Add '2.44140625' to each side of the equation. -3.125t + 2.44140625 + t2 = -1.5 + 2.44140625 Reorder the terms: 2.44140625 + -3.125t + t2 = -1.5 + 2.44140625 Combine like terms: -1.5 + 2.44140625 = 0.94140625 2.44140625 + -3.125t + t2 = 0.94140625 Factor a perfect square on the left side: (t + -1.5625)(t + -1.5625) = 0.94140625 Calculate the square root of the right side: 0.970260919 Break this problem into two subproblems by setting (t + -1.5625) equal to 0.970260919 and -0.970260919.Subproblem 1
t + -1.5625 = 0.970260919 Simplifying t + -1.5625 = 0.970260919 Reorder the terms: -1.5625 + t = 0.970260919 Solving -1.5625 + t = 0.970260919 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.5625' to each side of the equation. -1.5625 + 1.5625 + t = 0.970260919 + 1.5625 Combine like terms: -1.5625 + 1.5625 = 0.0000 0.0000 + t = 0.970260919 + 1.5625 t = 0.970260919 + 1.5625 Combine like terms: 0.970260919 + 1.5625 = 2.532760919 t = 2.532760919 Simplifying t = 2.532760919Subproblem 2
t + -1.5625 = -0.970260919 Simplifying t + -1.5625 = -0.970260919 Reorder the terms: -1.5625 + t = -0.970260919 Solving -1.5625 + t = -0.970260919 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.5625' to each side of the equation. -1.5625 + 1.5625 + t = -0.970260919 + 1.5625 Combine like terms: -1.5625 + 1.5625 = 0.0000 0.0000 + t = -0.970260919 + 1.5625 t = -0.970260919 + 1.5625 Combine like terms: -0.970260919 + 1.5625 = 0.592239081 t = 0.592239081 Simplifying t = 0.592239081Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.532760919, 0.592239081}Solution
t = {2.532760919, 0.592239081}
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