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Simplifying 0 = -16t2 + 128t + -32 Reorder the terms: 0 = -32 + 128t + -16t2 Solving 0 = -32 + 128t + -16t2 Solving for variable 't'. Combine like terms: 0 + 32 = 32 32 + -128t + 16t2 = -32 + 128t + -16t2 + 32 + -128t + 16t2 Reorder the terms: 32 + -128t + 16t2 = -32 + 32 + 128t + -128t + -16t2 + 16t2 Combine like terms: -32 + 32 = 0 32 + -128t + 16t2 = 0 + 128t + -128t + -16t2 + 16t2 32 + -128t + 16t2 = 128t + -128t + -16t2 + 16t2 Combine like terms: 128t + -128t = 0 32 + -128t + 16t2 = 0 + -16t2 + 16t2 32 + -128t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 32 + -128t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '16'. 16(2 + -8t + t2) = 0 Ignore the factor 16.Subproblem 1
Set the factor '(2 + -8t + t2)' equal to zero and attempt to solve: Simplifying 2 + -8t + t2 = 0 Solving 2 + -8t + t2 = 0 Begin completing the square. Move the constant term to the right: Add '-2' to each side of the equation. 2 + -8t + -2 + t2 = 0 + -2 Reorder the terms: 2 + -2 + -8t + t2 = 0 + -2 Combine like terms: 2 + -2 = 0 0 + -8t + t2 = 0 + -2 -8t + t2 = 0 + -2 Combine like terms: 0 + -2 = -2 -8t + t2 = -2 The t term is -8t. Take half its coefficient (-4). Square it (16) and add it to both sides. Add '16' to each side of the equation. -8t + 16 + t2 = -2 + 16 Reorder the terms: 16 + -8t + t2 = -2 + 16 Combine like terms: -2 + 16 = 14 16 + -8t + t2 = 14 Factor a perfect square on the left side: (t + -4)(t + -4) = 14 Calculate the square root of the right side: 3.741657387 Break this problem into two subproblems by setting (t + -4) equal to 3.741657387 and -3.741657387.Subproblem 1
t + -4 = 3.741657387 Simplifying t + -4 = 3.741657387 Reorder the terms: -4 + t = 3.741657387 Solving -4 + t = 3.741657387 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + t = 3.741657387 + 4 Combine like terms: -4 + 4 = 0 0 + t = 3.741657387 + 4 t = 3.741657387 + 4 Combine like terms: 3.741657387 + 4 = 7.741657387 t = 7.741657387 Simplifying t = 7.741657387Subproblem 2
t + -4 = -3.741657387 Simplifying t + -4 = -3.741657387 Reorder the terms: -4 + t = -3.741657387 Solving -4 + t = -3.741657387 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + t = -3.741657387 + 4 Combine like terms: -4 + 4 = 0 0 + t = -3.741657387 + 4 t = -3.741657387 + 4 Combine like terms: -3.741657387 + 4 = 0.258342613 t = 0.258342613 Simplifying t = 0.258342613Solution
The solution to the problem is based on the solutions from the subproblems. t = {7.741657387, 0.258342613}Solution
t = {7.741657387, 0.258342613}
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