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Simplifying 128 = -16t2 + 128t Reorder the terms: 128 = 128t + -16t2 Solving 128 = 128t + -16t2 Solving for variable 't'. Reorder the terms: 128 + -128t + 16t2 = 128t + -128t + -16t2 + 16t2 Combine like terms: 128t + -128t = 0 128 + -128t + 16t2 = 0 + -16t2 + 16t2 128 + -128t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 128 + -128t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '16'. 16(8 + -8t + t2) = 0 Ignore the factor 16.Subproblem 1
Set the factor '(8 + -8t + t2)' equal to zero and attempt to solve: Simplifying 8 + -8t + t2 = 0 Solving 8 + -8t + t2 = 0 Begin completing the square. Move the constant term to the right: Add '-8' to each side of the equation. 8 + -8t + -8 + t2 = 0 + -8 Reorder the terms: 8 + -8 + -8t + t2 = 0 + -8 Combine like terms: 8 + -8 = 0 0 + -8t + t2 = 0 + -8 -8t + t2 = 0 + -8 Combine like terms: 0 + -8 = -8 -8t + t2 = -8 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 = -8 + 16 Reorder the terms: 16 + -8t + t2 = -8 + 16 Combine like terms: -8 + 16 = 8 16 + -8t + t2 = 8 Factor a perfect square on the left side: (t + -4)(t + -4) = 8 Calculate the square root of the right side: 2.828427125 Break this problem into two subproblems by setting (t + -4) equal to 2.828427125 and -2.828427125.Subproblem 1
t + -4 = 2.828427125 Simplifying t + -4 = 2.828427125 Reorder the terms: -4 + t = 2.828427125 Solving -4 + t = 2.828427125 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 = 2.828427125 + 4 Combine like terms: -4 + 4 = 0 0 + t = 2.828427125 + 4 t = 2.828427125 + 4 Combine like terms: 2.828427125 + 4 = 6.828427125 t = 6.828427125 Simplifying t = 6.828427125Subproblem 2
t + -4 = -2.828427125 Simplifying t + -4 = -2.828427125 Reorder the terms: -4 + t = -2.828427125 Solving -4 + t = -2.828427125 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 = -2.828427125 + 4 Combine like terms: -4 + 4 = 0 0 + t = -2.828427125 + 4 t = -2.828427125 + 4 Combine like terms: -2.828427125 + 4 = 1.171572875 t = 1.171572875 Simplifying t = 1.171572875Solution
The solution to the problem is based on the solutions from the subproblems. t = {6.828427125, 1.171572875}Solution
t = {6.828427125, 1.171572875}
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